Quantum Painleve-Calogero Correspondence

Quan tum P ainlev ´ e-Calogero Corresp ondence A. Zabro din ∗ A. Zotov † July 2011 ITEP-TH-23/11 Abstract The P ainlev ´ e-Caloge ro corresp ondence is e xtended to auxiliary linear problems asso ciat ed with P ainlev ´ e equations. The linea r problems a re represente d in a new form whic h has a suggestiv e in terpretation as a “quant ized” version o f th e P ainlev ´ e- Calogero corresp ondence. Namely , the linear problem r esp onsible for the time ev olution is brough t in to the form of non-stationary Sc hr¨ odinger equation in imag- inary time, ∂ t ψ = ( 1 2 ∂ 2 x + V ( x, t )) ψ , whose Hamiltonian is a natur al quanti zation of the classical Calogero-lik e Hamiltonian H = 1 2 p 2 + V ( x, t ) for the corresp onding P ainlev ´ e equ ati on. Con ten ts 1 In tro duction 3 2 The general sc heme 7 2.1 Linear problems and compatibility conditions . . . . . . . . . . . . . . . 7 2.2 Ordinary second-order diﬀerential equation . . . . . . . . . . . . . . . . . 9 2.3 Non-stationary Sc hr¨ odinger equation . . . . . . . . . . . . . . . . . . . . 10 2.4 The linear problems and quantum Painle v ´ e-Calogero corresp ondence . . 11 3 P ainlev ´ e I 14 3.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Linearization and classical-quan tum corres p ondence for P I . . . . . . . . 15 4 P ainlev ´ e I I 16 4.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Linearization and classical-quan tum corres p ondence for P II . . . . . . . . 17 ∗ Institute of Bio chemical Physics, Kosy g ina str. 4, 119 991 Mo sco w, Russia a nd ITEP , B ol. Cher e- m ushkinsk ay a str. 25, 117 259 Mo sco w, Russia, E -mail: zabrodin@itep.r u † ITEP , Bol. Cheremushkinsk ay a str. 25, 1172 59 Mosc o w, Russia, E - mail: zotov@itep.ru 1 5 P ainlev ´ e IV 18 5.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Linearization and classical-quan tum corresp ondence for P IV . . . . . . . . 19 6 P ainlev ´ e I I I 21 6.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 The U – V pairs for P II I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.2.1 The case of the truncated P II I equation . . . . . . . . . . . . . . . 22 6.2.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 6.3 Classical-quan tum corresp ondence for P II I . . . . . . . . . . . . . . . . . 24 6.3.1 The case of truncated P II I equation . . . . . . . . . . . . . . . . . 24 6.3.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 7 P ainlev ´ e V 26 7.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7.2 The zero curv ature represen tation of the P V equation . . . . . . . . . . . 27 7.2.1 The mo diﬁed Jimbo-Miwa U – V pa ir for P V . . . . . . . . . . . . 27 7.2.2 Hyp erbolic parametrizatio n . . . . . . . . . . . . . . . . . . . . . 29 7.3 Classical-quan tum corresp ondence for P V . . . . . . . . . . . . . . . . . . 29 8 P ainlev ´ e VI 31 8.1 The equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.2 The zero curv ature represen tation of the P VI equation . . . . . . . . . . . 32 8.2.1 The mo diﬁed Jimbo-Miwa U – V pa ir for P VI . . . . . . . . . . . . 32 8.2.2 Elliptic parametrization . . . . . . . . . . . . . . . . . . . . . . . 34 8.3 Classical-quan tum corresp ondence for P VI . . . . . . . . . . . . . . . . . 35 9 Concluding remarks 40 App endix A 41 App endix B 44 App endix C 49 References 51 2 1 In tro duct ion The famous six nonlinear ordinary second-order diﬀeren tial equations disco v ered b y P .P ainlev ´ e, R.F uch s and B.Gam bier [1 , 2, 3] in the b eginning of the XX century are no w ada ys kno wn as t he P ainlev ´ e equations I–VI (P I –P VI ). Since that time they w ere extensiv ely studied and they still remain to b e among the most imp orta nt and most in teresting diﬀeren tial equations in mathematics and mat hematical ph ysics [4, 5]. Their applications include self-similar reductions of no n-linear in tegrable partial diﬀeren tial equations [6], correlation functions of integrable mo dels [7, 8], quan tum gra vit y and string theory [9], to polo gical ﬁeld theories [13], 2D p olymers [10], r a ndom matrices [1 1, 12] and sto c ha stic growth pro cesses [1 4 ], to men tion only few applications and few references. The idea t o asso ciate a system of line ar diﬀerential equations with each P ainlev ´ e equation go es ba c k to the seminal w ork b y R.F uchs [2]. In fact the theory of Painle v ´ e equations is in trinsically related to the mono dromy prop erties of linear ordinary diﬀer- en tial equations with rational co eﬃcien ts. Remark ably , the equations from the P ainlev ´ e list describe mono dromy preserving deformations of linear diﬀerential equations with es- sen tia l singularities. The classical references on t he sub ject are [15, 16 ]. The mono dromy approac h w as further deve lop ed b y H.Flasc hk a and A.New ell [6] and by M.Jim b o, T.Miw a and K.Ueno in the series of w orks [1 7, 18, 19 ], see also b o ok [20]. At presen t diﬀeren t t yp es of linear problems (scalar [2, 15], 2 × 2-ma t r ix [18] or 3 × 3-matrix [21 ]) are know n to b e asso ciated with P ainlev ´ e equations. The Hamiltonian theory of the Painle v ´ e equations is da ted bac k to the work [2 2] (for the mo dern dev elopmen ts and the extension to general Sc hlesinger systems see [23]). It turns out that all the six equations ha v e a Hamiltonian structure with time-dep enden t Hamiltonian f unctions whic h are p olynomials in the dep enden t v aria ble (the co ordinate) and suitably c hosen conjugate momen tum. They a re referred to as Ok a moto’s Hamilto- nians [2 4]. How ev er, the Ok amoto’s Hamiltonians fo r P II –P VI equations are o f a more complicated form than j ust momen tum squared plus p oten tial. This mak es a direct in- terpretation of P ainlev ´ e equations as classical mec hanical systems (a p oin t-lik e part icle on the line moving in a time-dep enden t p oten tial) problematic. Nev ertheless, suc h an in terpretation app ears to b e p ossible after a non-trivial canonical transformatio n which accomplishes the Painlev´ e-Calo ger o c orr esp ondenc e . The phenomenon kno wn in the lit era t ure as the (classical) P ainlev ´ e-Calogero cor- resp ondence [25] consists in t he p ossibilit y to represen t , by means of explicitly kno wn transformations of the dep enden t and indep enden t v ariables, all the six P ainlev ´ e equa- tions as no n- autonomous Hamiltonian systems ∂ t x = ∂ H ∂ p , ∂ t p = − ∂ H ∂ x with the standard one-particle Hamiltonian of the canonical f orm H = p 2 / 2 + V ( x, t ) for some p oten tial V ( x, t ) whic h explicitly dep ends on time t . In the case of P VI this Hamiltonian system resem bles the elliptic Calogero mo del with 2 particles in the cen ter of mass co ordinat es, whence the name P ainlev ´ e-Calo gero corresp ondence. (T o b e more precise, the P VI equation is a non-autono mous v ersion of a sp ecial rank-one case of the Inozem t sev’s extension [26 ] of the elliptic Calogero mo del.) F or the P VI equation this remark able observ a t io n w as made by Y u.Manin [27] who reviv ed the almost forgotten 3 w ork b y Painlev ´ e himself [28]. Later, K.T ak a saki [2 9] extended t his result to the other equations from the Painlev ´ e list. In principle, this extension can b e a c hiev ed b y a sp ecial degeneration pro cess from P VI to the lo w er mem b ers o f the Painlev ´ e family . Although the resulting Hamiltonia n systems hardly resem ble any Calogero- lik e mo dels, the name “P ainlev ´ e-Calogero corresp ondence” has b een extended to these cases as w ell. This also suggests generalizations to higher r a nk systems whic h w ere studied in [29]. The explicit form of the canonical transformations from the Ok a mo t o ’s Hamilto nia n systems to Calogero-like ones w as found in [29]. Here w e need only the co ordinate part of this transformat io n whic h is describ ed by the following theorem. Theorem 1 [29]. F or any of the six e quations fr o m the Pa inlev ´ e list written for a variable y ( T ) ther e exists a chang e of variables ( y , T ) → ( u , t ) of the form y = y ( u, t ) , T = T ( t ) that maps the Painlev´ e e quation to a se c ond-or der diﬀer ential e quation of the form ¨ u = − ∂ u V ( u, t ) (1.1) which is e quivalent to a non- a utonom ous Hamiltonian system ˙ u = ∂ H ( p, u, t ) /∂ p , ˙ p = − ∂ H ( p, u, t ) /∂ u with the Ham i ltonian H ( p, u, t ) = p 2 2 + V ( u , t ) (1.2) wher e V ( u , t ) is a time-dep e n dent p otential written in terms of r ational, h yp erb olic or el lip tic f unction s. This statemen t w as pro v ed in [29] by giving explicit form ulas for the correspo nding c hanges of v aria bles (see the table b elo w). W e call (1.1) the Calo ger o form of the P ainlev ´ e equation. The aim of this pap er is to extend the P ainlev ´ e-Calogero corresp ondence t o the linear problems asso ciated with the P ainlev ´ e equations. In fact w e suggest a new fo rm of the linear problems whic h allow s us to in terpret it as a “quan tized” v ersion of the P ainlev ´ e- Calogero corresp ondence. In other words, linearizatio n, i.e., going to the asso ciated linear problems, app ears to b e equiv alent to quantization of the Painlev ´ e equations regarded as classical mec hanical systems. The starting p oint is a system of t w o ﬁrst-order linear partial diﬀeren tial equations (PDE) in tw o v a r iables for a 2-comp onen t v ector- function ( ψ 1 , ψ 2 ) t of t he form presen ted, for example, in [18]. The t w o v ariables are the sp ectral parameter and the deformation pa- rameter. As is w ell kno wn, compatibilit y of the system is equiv alen t to the zero curv ature condition for the connection represen ted b y 2 × 2 matrices dep ending on the t w o v ariables. The next step is the c hange of the dep enden t and indep enden t v aria bles that leads to the Calogero-lik e form of the Painlev ´ e equations, supplemen ted b y a suitable change of the sp ectral parameter (p olynomial for P I , P II , P IV , exp onen tial for P II I , hyperb olic f or P V and elliptic for P VI ). A t this step the sp ectral pa rameter and the deformat ion para meter acquire the meaning of t he co ordinate and time v ariables for a no n-autonomous dynami- cal system with one degree of freedom. After an a dditional diag onal gauge tr a nsformation of a sp ecial form, the linear problems transformed in this w a y should b e rewritten a s a pair of tw o compatible linear PDE’s for a scalar ψ -function ψ = ψ 1 (the ﬁrst comp onen t of the v ector function). One of them is an ordinary second-order diﬀeren tial equation 4 with co eﬃcien ts explicitly depending on time a nd on the dep enden t v ariable. After a simple transformation of the ψ -function the term with the ﬁrst order deriv ativ e cancels, and one o btains a stationary Sc hr¨ odinger equation with a p oten tial function whic h de- p ends on time in b oth explicit and implicit w a ys, with the implicit dep endenc e coming from the dep enden t v ariable. T he isomono dromy pr o blem f o r this equation, i.e., time- dep enden t deformation of the p otential preserving the mono dromy of solutions, is kno wn to b e equiv alen t to the Painle v ´ e equation. The k ey new elemen t introduced in this pa per is the second equation of the pair, the one describing the time evolution. W e sho w that f or all the six P ainlev ´ e equations (and an y v alues of the standard parameters α , β , γ , δ in volv ed) it can b e represen ted in the form of the non-stationary Schr¨ odinger e quation in imaginar y time, ∂ t Ψ = ˆ H Ψ , whose Hamiltonian is the standard 1D Sc hr¨ odinger op erator ˆ H = 1 2 ∂ 2 x + V ( x, t ) whic h is a natur al quantization of the classic al Calo ger o-like Ham i l toni a n asso ciated with the P ainlev ´ e equation at hand (to b e mo r e precise, fo r P VI the parameters α, β , γ , δ in the quan tized Hamiltonian app ear to b e shifted by “quan tum corrections” ± 1 8 , similar shifts of some of the parameters tak e place also fo r P V and P IV ). Therein lies the quan- tum P a inlev ´ e-Calogero corresp ondence, or a classical-quan tum corresp ondence for the P ainlev ´ e equations. Indeed, on the Calog ero side, one now has a quantum Calog ero - lik e or Inozem tsev mo del in a non-stationary state describ ed by the w av e f unction Ψ whic h diﬀers fr o m ψ by a co ordinate-indep enden t factor. On the P ainlev ´ e side, this Ψ-function is a common solution to the linear problems asso ciated with the P ainlev ´ e equation. So- lutions of the P ainlev ´ e equation itself can b e extracted from the asymptotic b ehav ior of the Ψ-function near singular p oints . The main results o f this w ork are summarized in the f ollo wing “quantum” v ersion of Theorem 1: Theorem 2. F or any of the six e quations fr om the Painlev´ e list w ritten in the C alo ger o form ( 1 .1) as classic al Hamiltonia n systems with time-d e p endent Hamiltonia ns H ( p, u , t ) (1.2) ther e e x ists a p air of c omp atible line ar pr oble m s      ∂ x Ψ = U ( x, t, u, ˙ u, { c k } ) Ψ ∂ t Ψ = V ( x, t, u, ˙ u, { c k } ) Ψ , Ψ = ψ 1 ψ 2 ! , (1.3) wher e U and V ar e sl 2 -value d functions, x is a sp e ctr al p ar ameter, t is the time variable and { c k } = { α, β , γ , δ } is the set of p a r ameters involve d in the Painlev´ e e quation, such that 1) The zer o curvatur e c o ndition ∂ t U − ∂ x V + [ U , V ] = 0 (1.4) is e quivalent to the Painlev´ e e quation (1.1) for the variable u deﬁne d as any (simple) zer o of the right upp er element of the matrix U ( x, t ) in the sp e ctr al p ar ameter: U 12 ( u, t ) = 0 ; 5 2) The function Ψ = e R t H ( ˙ u,u, t ′ ) dt ′ ψ 1 wher e ψ 1 is the ﬁrst c omp one nt of Ψ satisﬁes the non-stationary Schr¨ oding e r e quation in imaginary time ∂ t Ψ =  1 2 ∂ 2 x + ˜ V ( x, t )  Ψ (1.5) with the p otential ˜ V ( x, t ) = V ( x, t, { ˜ c k } ) = 1 2 h det( U ) − ∂ x U 11 + 2 V 11 i which c oinc i d es with the classic al p otential V ( x, t ) = V ( x, t, { c k } ) up to p ossible shifts of the p ar ameters { c k } : ( ˜ α, ˜ β ) = ( α , β + 1 2 ) for P IV , ( ˜ α, ˜ β , ˜ γ , ˜ δ ) = ( α − 1 8 , β + 1 8 , γ , δ ) for P V ( ˜ α, ˜ β , ˜ γ , ˜ δ ) = ( α − 1 8 , β + 1 8 , γ − 1 8 , δ + 1 8 ) for P VI . F or reader’s con v enience w e collect the c hang es of v ar ia bles f rom the orig inal y , T to u, t required f o r passing to the Calogero form and the corresp onding c hang e of the sp ectral parameter from ratio nal one, X , to x , in the follo wing table: Equation y ( u, t ) T ( t ) X ( x, t ) U 12 ( x, t ) P I u t x x − u P II u t x x − u P IV u 2 t x 2 x 2 − u 2 P II I e 2 u e t e 2 x 2 e t/ 2 sinh( x − u ) P V coth 2 u e 2 t cosh 2 x 2 e t sinh( x − u ) sinh( x + u ) P VI ℘ ( u ) − ℘ ( ω 1 ) ℘ ( ω 2 ) − ℘ ( ω 1 ) ℘ ( ω 3 ) − ℘ ( ω 1 ) ℘ ( ω 2 ) − ℘ ( ω 1 ) ℘ ( x ) − ℘ ( ω 1 ) ℘ ( ω 2 ) − ℘ ( ω 1 ) ϑ 1 ( x − u ) ϑ 1 ( x + u ) h ( u , t ) In the last column the right upp er elemen t of the mat r ix U ( x, t ) is give n. One can see that in all cases u is indeed a simple zero of U 12 ( x, t ). The function h ( u, t ) is some function of u , t only to b e sp eciﬁed in Section 8. The W eierstrass ℘ -function ℘ ( z ) = ℘ ( z | 1 , τ ) and the Jacobi t heta- function ϑ 1 ( x ) = ϑ 1 ( x | τ ) in the last line of the table dep end on t in a no n- trivial wa y throug h the second p erio d τ = 2 π it . The half -p eriods are deﬁned as ω 1 = 1 2 , ω 2 = 1 2 (1 + τ ), ω 3 = 1 2 τ . When this work w as completed, w e were informed b y B.Suleimano v that he realized the role o f non-stationary Schr¨ odinger-lik e equation in linear problems for P ainlev ´ e equa- tions bac k in 1994 a nd obta ined similar results [30]. In distinction to o ur approac h, he starts with the scalar linear problems of the F uc hs-Garnier t yp e with rational sp ectral pa- rameter [2, 15] and sho ws that their compatibility implies yet a nother linear equation for the same wa ve function, whic h is of the non- stationary Schr¨ odinger form, with quantum 6 Hamiltonian b eing a quantization of the corresp onding Ok amoto’s Hamiltonian. The precise connection b et w een the t w o approaches deserv es further elucidation. The presen tation is organized in suc h a w a y that eac h Painlev ´ e equation is discussed in a separate section, in the order of increasing complexit y , fr o m P I to P VI (Sections 3–8). W e tried to make eac h section self-contained, so that they could b e read indep enden tly of eac h other. Ho w ev er, eac h section contains references to Section 2, where the g eneral construction is o ut lined. Note that in our list P IV stands b efore P II I b ecause in a certain sense the complex ity of the latter exceeds that of the former. This is due to the fact that the P I , P II and P IV equations need ratio nal parametrization to b e represen ted in the Calogero-lik e f orm while P II I and P V require exp onen tial and h yp erb olic parametrizations for that purp ose. The highest mem b er, P VI , is the most complicated ob ject. It requires parametrization in terms of elliptic functions. One can see that the calculations whic h are necessary to prov e Theorem 2 and to v erify the classical-quan tum corr esp ondence, b eing really short and t ransparen t fo r P I , b ecome ve ry lo ng and tedious for P VI . In the case of P VI (and to some exten t of P V ), the situation is ag g ra v ated b y the fact that neither the change of the sp ectral para meter nor the gauge transformation are kno wn from the v ery b eginning and should b e either guessed or f o und by solving a diﬀeren tial equation. The three app endices a r e all related to the P VI equation. In App endix A some details of explicit v eriﬁcation of the zero curv ature conditio n are give n. App endix B con tains the necessary information on theta- functions and elliptic functions. In App endix C the sp ecial dia gonal gauge transformatio n together with the c hange of the sp ectral parameter for t he linear problems for the P VI equation is deriv ed. 2 The general sc heme 2.1 Linear problems and compatibilit y conditions As is kno wn, any P ainlev ´ e equation I-VI can b e r epresen ted as the compatibility condition for a pair of linear problems dep ending on a sp ectral parameter. W e need the linear problems suc h that they lead directly to the P ainlev ´ e equations in the Calogero form. They can b e obtained from the linear problems with rational sp ectral parameter by a prop er c hange of v ariables. T he existence o f suc h a change of v ariables will b e prov ed separately for eac h equation P I -P VI b y an explicit calculation. No w suppo se that w e are giv en with suc h a pair of linear problems:      ∂ x Ψ = U ( x, t ) Ψ ∂ t Ψ = V ( x, t ) Ψ , Ψ = ψ 1 ψ 2 ! , (2.1) where the 2 × 2 matrices U , V explicitly dep end on the sp ectral parameter x (whic h in our approa c h has the meaning of co o rdinate), on the deformation parameter t (whic h in our approach has the meaning of time) and con tain an unkno wn functions of t to b e constrained by the condition that the tw o equations hav e a f amily o f common solutions. This function is going to satisfy one of the six P ainlev ´ e equations (in the Calogero form). In fact t he latter is equiv alen t to the compatibility of the linear pr o blems expressed as 7 the zero curv at ure equation (integrabilit y condition) ∂ x V − ∂ t U + [ V , U ] = 0 . (2.2) Set U = a b c d ! , V = A B C D ! . Our matrices U , V will b e alwa ys traceless, i.e., a + d = 0, A + D = 0. In this nota tion, the zero curv at ure equation yields:                a t − A x + bC − cB = 0 b t − B x + 2 aB − 2 bA = 0 c t − C x + 2 cA − 2 aC = 0 . (2.3) Here and b elo w a t , A x , etc mean partial deriv ative s with resp ect to t, x . T o av oid a misunderstanding, w e emphasize that the time v ariable t en ters the matrix elemen ts in t w o w a ys: explicit a nd implicit. The latter means the time dep endence through the unkno wn functions of t (dep enden t v ariables). The notation a t , etc implies the full time diﬀeren tiating whic h takes into a ccount t he time dep endence of t he b oth t yp es. The function that satisﬁes the Painlev ´ e equation in the Calog ero fo r m will b e denoted b y u = u ( t ). It can b e deﬁned a s zero of the righ t upp er elemen t of the matrix U ( x, t ) as a function of the sp ectral par a meter x : b ( u ) = 0. W e will see that this zero is alwa ys of the ﬁrst order a nd diﬀeren t p ossible c hoices (in the case when the function b ( x ) has more than one zero in a suitably c hosen fundamen tal domain) lead to the same equation. It is imp ortan t that the matrix functions U ( x, t ) , V ( x, t ) ha v e p oles in x a t the p oin ts whic h ma y dep end o n time but not thr ough the dep endent varia ble u . In fact for P I – P V equations they are time indep enden t while fo r the P VI equation tw o p oles are ﬁxed and other t w o linearly dep end on the time v ariable. In what fo llo ws w e will c ho ose the matrices U , V suc h that b x = 2 B . (2.4) (the meaning and adv antages of this condition will b e clear lat er). Given any t w o matrix functions U , V , this equalit y can b e a lw ays attained b y means o f a suitable diagonal gauge transformatio n of the linear system ( 2 .1) (see b elo w). In principle, one can then exclude A and C from the zero curv ature equations (2.3) and obt a in a functional r elat io n for a, b and c but we will not follo w this route here. Let us only men t io n, for future reference, that if the zero curv ature equation and the condition (2.4) a re imp osed, then A is expressed through a and b as fo llo ws: 2 A = b t + ab x b − b xx 2 b . (2.5) The system (2.1) a dmits gauge t r ansformations ˜ Ψ = Ω Ψ with a matrix Ω whic h can dep end o n x, t . The gauge transformed system has t he same form      ∂ x ˜ Ψ = ˜ U ( x, t ) ˜ Ψ ∂ t ˜ Ψ = ˜ V ( x, t ) ˜ Ψ (2.6) 8 with ˜ U = Ω − 1 U Ω − Ω − 1 ∂ x Ω , ˜ V = Ω − 1 V Ω − Ω − 1 ∂ t Ω . (2.7) In the next sections this transformation will b e applied in t he o pposite direction, from matrices ˜ U , ˜ V obtained at an in termediate stage o f calculations to matrices U , V in the ﬁnal form. This is equiv alent to applying the inv erse tra nsfor mat ion. In particular, let Ω = ω 0 0 ω − 1 ! (2.8) b e a diagonal matrix, then U =    ˜ a + ∂ x log ω ˜ bω 2 ˜ cω − 2 ˜ d − ∂ x log ω    , V =    ˜ A + ∂ t log ω ˜ B ω 2 ˜ C ω − 2 ˜ D − ∂ t log ω    . (2.9) Let us consider the tw o linear problems (2.1) in detail. Explicitly , w e hav e:      ∂ x ψ 1 = aψ 1 + bψ 2 ∂ x ψ 2 = cψ 1 + dψ 2 ,      ∂ t ψ 1 = Aψ 1 + B ψ 2 ∂ t ψ 2 = C ψ 1 + D ψ 2 Applying ∂ x to the ﬁrst equation of the ﬁrst system, and using the second equation, w e obtain ∂ 2 x ψ 1 − ( a + d ) ∂ x ψ 1 + ( ad − bc ) ψ 1 − ( a x ψ 1 + b x ψ 2 ) = 0 . (2.10) Using the linear equations ab ov e, one can express ψ 2 through ψ 1 in t w o diﬀeren t w ay s: ψ 2 = ∂ x ψ 1 − aψ 1 b = ∂ t ψ 1 − Aψ 1 B . (2.11) The ﬁrst p ossibilit y leads t o a closed o r dinary second-order diﬀeren tial equation for ψ 1 while the second o ne leads to a partial diﬀerential equation for ψ 1 as a function of x, t . As w e shall see so on, b oth ha v e the form of Sc hr¨ odinger equations, statio nary and non- stationary . This pair of scalar equations is equiv alen t to the origina l system (2.1) in the sense that their compatibilit y implies Painle v ´ e equations for the dep enden t v ariable. One can also sa y that the second equation describ es isomono dromic deformations of the ﬁrst one. Let us consider them separately . F rom now o n w e will write simply ψ instead o f ψ 1 . 2.2 Ordinary second -order diﬀeren tial equ ation Using the ﬁrst equalit y in (2.11), we get an o r dina r y second-order diﬀerential equation for ψ := ψ 1 : ∂ 2 x ψ −  a + d + b x b  ∂ x ψ +  ad − bc − a x + b x a b  ψ = 0 . The co eﬃcien t functions here are expressed through en tries of the matrix U ( x, t ). F or traceless mat rices with the condition (2.4) the equation acquires the form ∂ 2 x ψ − b x b ∂ x ψ +  ad − bc − a x + 2 A − b t b + b xx 2 b  ψ = 0 9 (here fo r the transformation of the last term (2 .5) has b een used) or 1 2 ∂ 2 x − b x 2 b ∂ x + W ( x ) ! ψ = 0 , (2.12) where W ( x ) = 1 2 ( ad − bc − a x + 2 A ) − 1 2 b  − ∂ t + 1 2 ∂ 2 x  b. (2.13) The substitution ψ = √ b ˇ ψ kills the ﬁrst deriv ative term in eq. (2.12) and brings it to the form of stationary Sc hr¨ odinger equation  1 2 ∂ 2 x + ˇ W ( x )  ˇ ψ = 0 (2.14) with the p otential ˇ W = W + 1 4 ∂ 2 x log b − 1 8 ( ∂ x log b ) 2 . (2.15) This equation has for mal solutions with the WKB-lik e asymptotes near p o les of the p oten tial: ˇ ψ ( x ) ∼ = ( − 2 ˇ W ) − 1 4 e ± R x √ − 2 ˇ W dx ′ . (2.16) An expansion o f the right hand side near singularities of the p oten tial allow s one to extract solutio ns to the corresp onding Painlev ´ e equation. 2.3 Non-stationary Sc hr¨ odinger equation The second p ossibilit y in (2.11) is more in teresting for us here. It leads t o a partial diﬀeren tial equation for ψ = ψ 1 as a function of x, t : ∂ 2 x ψ − ( a + d ) ∂ x ψ + ( ad − bc ) ψ −  a x − b x A B  ψ − b x B ∂ t ψ = 0 . (2.1 7) The co eﬃcien t f unctions here are expressed t hrough entries of the b oth matrices U ( x, t ), V ( x, t ). F or traceless matrices with the condition (2.4) the equation simpliﬁes: ∂ 2 x ψ +  ad − bc − a x + 2 A  ψ − 2 ∂ t ψ = 0 . (2.18) The role of the condition (2.4) is thus to make constan t the co eﬃcien t in fro nt of t he time deriv a tiv e ( t he sp eciﬁc v alue 2 of the constant is just a matter o f normalizatio n) . Equation (2.18) is cen t r a l for what fo llo ws. Cle arly , it has the form of a non-stationa ry Sc hr¨ odinger equation in imaginary time: ∂ t ψ =  1 2 ∂ 2 x + U ( x, t )  ψ (2.19) with the p otential U ( x, t ) = 1 2 ( ad − bc − a x ) + A = 1 2 det U − a x 2 + A. (2.20) 10 In the subsequen t sections 3–8 w e v erify , b y means of the case study , tha t fo r all Painlev ´ e equations the dep enden t v ariable u en ters this p otential only thro ugh an irrelev a n t x - indep enden t term while x - dep enden t terms contain the time v ariable in t he explicit form only . Moreo v er, this p oten tial turns out to b e the same as the clas s ic al me chanic al p otential for Painlev´ e e quations written in the Calo ger o form . (T o b e precise, w e should p oin t out that f o r higher mem b ers of the P ainlev ´ e family , P IV – P VI , the co eﬃcien ts in fron t of diﬀeren t terms of the p oten tial may b e mo diﬁed). This pro vides the quantum v ersion of the Painlev ´ e-Calogero cor r espondence. Summing up, w e hav e reduced t he linear system (2.1 ) for the v ector function Ψ = ( ψ 1 , ψ 2 ) t to t w o scalar equations for ψ := ψ 1 :             1 2 ∂ 2 x − 1 2 ( ∂ x log b ) ∂ x + W ( x, t )  ψ = 0 ∂ t ψ =  1 2 ∂ 2 x + U ( x, t )  ψ . (2.21) The second equation describes isomono dromic deformations of the ﬁrst one and their compatibilit y implies the P ainlev ´ e equation (in the Calogero form) fo r the function u = u ( t ) deﬁned as a (simple) zero o f the function b ( x ): b ( u ) = 0. The x -dep enden t part of the p oten tial U ( x, t ) do es not contain the dep enden t v aria ble u . Note that the p otentials W and U are r elated b y W = U − 1 2 ∂ t log b + 1 4 ∂ 2 x log b + 1 4 ( ∂ x log b ) 2 , so the p otential W ( x, t ) has a n apparen t singularity at x = u ( t ). One can see that equations (2.21) imply the scalar linear pro blems in the fo r m sug- gested b y R.F uc hs [2] and R.Garnier [15]. Indeed, passing to the function ˇ ψ = ψ / √ b and com bining the t w o equations (2.21), one obtains the linear system             1 2 ∂ 2 x + ˇ W ( x, t )  ˇ ψ = 0 ∂ t ˇ ψ =  Λ ∂ x − 1 2 ( ∂ x Λ)  ˇ ψ , Λ := 1 2 ∂ x log b , (2.22) with ˇ W giv en b y (2 .1 5), which is exactly of the F uc hs-Garnier f orm. The inte grability condition for this system is ∂ t ˇ W = 2 ˇ W ∂ x Λ + Λ ∂ x ˇ W + 1 4 ∂ 3 x Λ . (2.23) 2.4 The linear problems and quan tum P ainlev ´ e-Calogero c or- resp ondence In t his subsection we giv e a general view on what w e are going to do in sections 3–8 for the pa rticular Painle v ´ e equations. In t he original form, the P ainlev ´ e equations can b e written as ∂ 2 T y = R ( T , y , ∂ T y ) , (2.24) 11 where R is a rational function of the indep enden t v ar iable T , the dependen t v ariable y and its T - deriv ativ e. The Painlev ´ e-Calogero corresp ondence means the existence o f a c hange of v ariables from y , T to x, t of the form y = y ( x, t ), T = T ( t ) suc h that eq. (2.24) in the new v ariables a cquire s the form ¨ x = − ∂ x V ( x, t ) (2.25) whic h is the Newton equation for motion of a p oint-lik e particle on the line in a time- dep enden t p oten tial V ( x, t ). In order to indicate the dep endence on the parameters α, β , γ , δ whic h ma y enter the P ainlev ´ e equations, w e will write V ( x, t ) = V ( α,β ,γ ,δ ) ( x, t ). As it w as already said in t he Intro duction, w e call (2 .25) the Ca l o ger o form of the Painlev ´ e equation. Hereafter, the dot means the t -deriv ative. It should b e noted that P I and P II equations are a lready of the Calogero form, so no c hange of t he v ariables is necessary , for P II I – P V equations the transformation y → x bring ing the equations to the Calog ero form do es not dep end on t , and o nly for P VI this t ransformation is actually t -dep enden t. The linear pro blems o f the necessary form describ ed in section 2.1 hav e b een kno wn for lo w er members o f the P ainlev ´ e family but not for higher ones (esp ecially for P V and P VI ). Therefore, w e should start from a known v ersion o f the linear problems and then transform it to the desired fo rm. A con v enien t starting p oin t is the pair of compatible linear problems      ∂ X Ψ = U ( X , T ) Ψ ∂ T Ψ = V ( X , T ) Ψ (2.26) for a tw o- component vec tor function Ψ , where the ma t r ices U ( X , T ), V ( X , T ) are rational functions of the spectral parameter X g iven in [18] for all the six P ainlev ´ e equations. The transformation from this pair o f matrices to the pair of matrices U ( x, t ), V ( x, t ) with the desired prop erties will b e done in tw o steps: { U ( X, T ) , V ( X, T ) } R − → { ˜ U ( x, t ) , ˜ V ( x, t ) } G − → { U ( x, t ) , V ( x, t ) } . The transformation R is a re-para metrizatio n of the time and sp ectral par a meter corresp onding to the change o f v ariables that prepares the Calog ero for m of the P ainlev ´ e equation from the o riginal o ne. Here a r e some general relations fo r a c hange of v ariables from X , T to x, t of the fo r m X = X ( x, t ), T = T ( t ). Clearly , suc h a c hange of v ariables implies the fo llo wing relatio ns for the partial deriv atives : ∂ x = ∂ X ∂ x ∂ X , ∂ t = ∂ X ∂ t ∂ X + ∂ T ∂ t ∂ T . This means that the linear problems (8.6) are transformed as follo ws:              ∂ x Ψ = ∂ X ∂ x UΨ ∂ t Ψ = ∂ T ∂ t V + ∂ X ∂ t U ! Ψ . (2.27) 12 Therefore, the U – V pair in t he v a riables x, t is ˜ U ( x, t ) = ∂ X ∂ x U ( X ( x, t ) , T ( t )) ˜ V ( x, t ) = ∂ T ∂ t V ( X ( x, t ) , T ( t )) + ∂ X ∂ t U ( X ( x, t ) , T ( t )) , (2.28) where the en tries of the matrices U , V in t he right hand side should b e expressed t hr o ugh the new v ariables x, t . Note that w e delib erately use the same letter x as in the equation of the classical motion (2.25) to stress the fact that it is this v ariable ( x -co ordinate of a particle on t he line) which is going to b e “ quan tized” in the “quantum” v ersion of the P ainlev ´ e-Calogero corresp ondence in the sense tha t the momen tum p = ˙ x is going to b e replaced b y the o p erator ∂ x . This a nalogy is justiﬁed b y the ﬁnal fo rm ulas. The zero curv at ure conditio n for the pair of matrices ˜ U ( x, t ), ˜ V ( x, t ) is equiv alen t to the Painlev ´ e equation in t he Calogero form ¨ u = − ∂ u V ( u, t ) (2.29) for the function u = u ( t ) deﬁned a s a (simple) zero o f the right upp er elemen t ˜ b ( x, t ) = ˜ U 12 ( x, t ) of the matrix ˜ U : ˜ b ( u, t ) = 0. (T o av oid a misunderstanding, w e should stress that the time dep endence o f the function u ( t ) is deﬁned n ot by this equation but b y the P ainlev ´ e equation.) In general, the so obtained ma t r ices ˜ U ( x, t ), ˜ V ( x, t ) do not o bey the condition (2.4). The tra nsformation G is a diagonal gauge transformation of the f orm (2.9) with a sp ecially adjusted function ω ( x, t ) suc h t ha t the condition (2.4) for the gauge-tr ansformed matrices is satisﬁed. Here an imp ortant remark is in order. Giv en an y t w o 2 × 2 matrix functions ˜ U ( x, t ), ˜ V ( x, t ), one can alw a ys ﬁnd a scalar function ω ( x, t ) suc h that the upp er right en tries of the ga ug e-transformed matrices, b = ˜ bω 2 = ˜ U 12 ω 2 , B = ˜ B ω 2 = ˜ V 12 ω 2 , are related b y t he equation b x = 2 B . Indeed, suc h a function ω can b e f ound as a solution to the diﬀeren tial equation ∂ x log ω = ˜ B ˜ b − 1 2 ∂ x log ˜ b . A non-trivial a dditional constraint on the function ω is tha t it should fa ctorize into a pro duct of t w o functions suc h that o ne of them dep ends on x, t but do es not c ontain the dep endent variable u and a no ther one dep ends on t only (through b oth dep enden t and indep enden t v ar ia bles). In fact t his is a necessary condition for the p erfect classical- quan tum correspo ndenc e. Otherwise the sp ectral parameter and the dep enden t v ariable ha v e no c hance t o separate in the p otential of the non- stat io nary Sc hr ¨ odinger equation. In fact the example of P VI sho ws that the tw o transforma t io ns, R and G , should b e found sim ultaneously from the condition that the function ω be of the sp ecial f o rm in whic h the dep enden t v a r ia ble separates from the sp ectral pa r ameter. The resulting pair of matrices U ( x, t ), V ( x, t ) is the one that w as discussed in section 2.1. The zero curv ature condition f or these matrices is equiv alen t to the P ainlev ´ e equation (2.29) for the function u = u ( t ) whic h can b e equiv alently deﬁned as a (simple) zero of the righ t upp er elemen t b ( x, t ) = U 12 ( x, t ) of the matrix U : b ( u, t ) = 0. One can also che c k that the v alue of t he diagonal elemen t, U 11 ( x, t ), at x = u is a canonically conjuga te 13 v ariable to u , in accordance with the general constructions of [33, 3 4]. (A more detailed discussion of t his p oint will b e give n elsewhere.) F urther, we are going to reduce the system of linear problems (2.1) to the pair of scalar Schr¨ odinger-lik e equations (2.21) a ccording to the pro cedure o utlined in sections 2.2 and 2.3. The result merits attention and further understanding from “ﬁrst principles”. The explicit calculations in eac h case sho w that fo r an y Painlev ´ e equation (with p ossible parameters α, β , γ , δ ) the follo wing holds true: • The v aria bles x, u separate in the non- stationary Sc hro dinger equation meaning that U ( x, t ) = V ( ˜ α, ˜ β , ˜ γ , ˜ δ ) ( x, t ) − H ( α,β ,γ ,δ ) ( ˙ u, u ) , (2.30) where the p o ten tial V ( ˜ α, ˜ β , ˜ γ , ˜ δ ) ( x, t ) is of the same form as t he one for the clas- sical equation (2 .25) (or (2.29)) with p ossibly mo diﬁed para meters and the x - indep enden t term, H ( α,β ,γ ,δ ) ( ˙ u, u ), is t he classical Hamiltonian H ( ˙ u, u ) = H ( α,β ,γ ,δ ) ( ˙ u, u ) = 1 2 ˙ u 2 + V ( α,β ,γ ,δ ) ( u, t ) for t he Painlev ´ e equation in the Calogero form; • F or P I –P II I the parameters in the quantum Hamiltonian are the same as in the classical one while for P IV –P VI some or all parameters should b e shifted: ( ˜ α, ˜ β ) = ( α, β + 1 2 ) f or P IV , ( ˜ α, ˜ β , ˜ γ , ˜ δ ) = ( α − 1 8 , β + 1 8 , γ , δ ) for P V and ( ˜ α, ˜ β , ˜ γ , ˜ δ ) = ( α − 1 8 , β + 1 8 , γ − 1 8 , δ + 1 8 ) for P VI . This means that the function Ψ( x, t ) = e R t H ( ˙ u,u ) dt ′ ψ ( x, t ) (2.31) is a common solution to the linear diﬀeren tial equations             1 2 ∂ 2 x − 1 2 ( ∂ x log b ) ∂ x + W ( x, t )  Ψ = 0 ∂ t Ψ =  1 2 ∂ 2 x + V ( x, t )  Ψ . (2.32) The second one is the non-stationar y Sc hr¨ odinger equation ∂ t Ψ = H ( ∂ x , x )Ψ whose Hamiltonian is the natural quan tization of the classical Hamiltonian of the P ainlev ´ e equation, p ossibly with mo diﬁed pa rameters (such a mo diﬁcation, if a n y , can b e re- garded as a “quan tum correction”). This is what w e call the quan tum Painlev ´ e-Calogero corresp ondence or the classical-quan tum corresp ondence for the Painlev ´ e equations. 3 P ainl e v ´ e I 3.1 The equ ation The P I equation 4 ¨ x = 6 x 2 + t (3.1) 14 is already of the Calogero form from the v ery b eginning, so no c hange of v ariables is necessary in this case. It can b e written in the standard Hamiltonian for m as ˙ x = ∂ H I ∂ p , ˙ p = − ∂ H I ∂ x with the classical time-dep enden t Hamilto nian H I = H I ( p, x ) = p 2 2 − x 3 2 − tx 4 . (3.2) One may introduce the p oten tial V I ( x ) = − x 3 2 − tx 4 , (3.3) then the P I equation tak es the form ¨ x = − ∂ x V I ( x ) whic h is the Newton equation for a p oin t-lik e particle on the line in the time-dep enden t p otential. Note that the partial and full time deriv a tiv es of the Hamiltonian coincide: ∂ H I ∂ t = dH I dt = − x ( t ) 4 (3.4) (the ﬁrst equalit y is o f course the general prop ert y of Hamiltonia ns for non- conserv ative systems while the second one is sp eciﬁc f or the P I equation). 3.2 Linearization and classical-quan tum c orresp ondence for P I In the case of P I the general construction o ut lined in section 2 is esp ecially simple and transparen t b ecause it do es not need neither the c hange of v aria bles nor the gaug e trans- formation. The P I equation 4 ¨ u = 6 u 2 + t is kno wn to b e the compatibility condition fo r the linear pro blems (2.1) with the matrices U ( x, t ) =    ˙ u x − u x 2 + xu + u 2 + 1 2 t − ˙ u    , V ( x, t ) =    0 1 2 1 2 x + u 0    (3.5) whic h are alr eady of the f orm implied in section 2.1. Note tha t u is the simple zero of the r ig h t upp er elemen t of the ma t r ix U ( x, t ): b ( u ) = 0. Another meaning of the P I equation (whic h w e will not discuss here) is the condition that the mono drom y data of the ﬁrst linear problem in (2.1) b e indep enden t of the parameter t . The sp ectral parameter is denoted b y x . W e delib erately use the same letter x as in the equation of t he classic al motion (3.1) to stress the fact that it is this v ariable ( x - co ordinate of a particle on the line) whic h is going to b e “quan tized” in the “quan tum” v ersion of the P ainlev ´ e-Calogero corresp ondence in the sense that the momen tum p = ˙ x is going to b e replaced b y the op erator ∂ x . The notatio n with the same idea in mind will b e used b elo w for other P ainlev ´ e equations. 15 It remains to apply the general form ulas of section 2 . Consider equation (2.19). In the case of P I ad − bc = − x 3 − tx 2 − ˙ u 2 + u 3 + tu 2 , a x = A = 0 . so the calculation of the p otential U ( x, t ) is v ery simple. As a r esult, w e o btain the non-stationary Sc hr¨ odinger equation (in imaginary time) ∂ t ψ = 1 2 ∂ 2 x − x 3 2 − tx 4 − H I ( ˙ u, u ) ! ψ , (3.6) where H I ( ˙ u, u ) is giv en by (3.2). W e can write it in the form ∂ t ψ =  H I ( ∂ x , x ) − H I ( ˙ u, u )  ψ (3.7) where H I ( ∂ x , x ) = 1 2 ∂ 2 x − x 3 2 − tx 4 is the quantum Hamiltonian op erator obtained as a literal quantization of the classical Hamiltonian (3.2). The function Ψ( x, t ) = e R t H I ( ˙ u,u ) dt ′ ψ ( x, t ) (3.8) th us ob eys the non-statio na ry Sc hr¨ odinger equation ∂ t Ψ = H I ( ∂ x , x )Ψ =  1 2 ∂ 2 x + V I ( x, t )  Ψ (3.9) without a free t -dep enden t term. T o conclude, w e ha v e t w o equiv alen t represen t a tions of the P I equation. One is a classical motion in the time-dep enden t cubic p oten tial with Hamiltonian (3.2). The co ordinate of the particle a s a function o f time ob eys the P I equation. Another repre- sen ta tion is a time-dep enden t quan tum mec hanical particle in the same time-dep enden t p oten tial. The non- stationary Sc hr¨ odinger equation fo r this quan tum system in the co- ordinate represen tation sim ultaneously serv es as the linear problem f or time ev olut io n asso ciated with the Painle v ´ e equation 1 . 4 P ainl e v ´ e I I 4.1 The equ ation The P II equation ¨ x = 2 x 3 + tx − α , (4.1) 1 As we learned from B.Suleimanov after completetion of this work, this fact was po in ted out in [30, 31], see also [32]. 16 where α is an arbitrary parameter, is already of the Calog ero form from t he v ery b egin- ning, so no c hange of v ariables is necessary in this case. It can b e written in the standard Hamiltonian form as ˙ x = ∂ H II ∂ p , ˙ p = − ∂ H II ∂ x with the classical time-dep enden t Hamilto nian H II = H II ( p, x ) = p 2 2 − 1 2  x 2 + t 2  2 + αx = p 2 2 − x 4 2 − tx 2 2 − t 2 8 + αx. (4.2) One may introduce the p oten tial V II ( x ) = − 1 2  x 2 + t 2  2 + αx, (4.3) then the P II equation takes the Newton for m ¨ x = − ∂ x V II ( x ). Note t ha t the partial a nd full time deriv a tiv es of the Hamiltonian coincide: ∂ H II ∂ t = dH II dt = − x 2 ( t ) 2 − t 4 (4.4) (again, the ﬁrst equality is a general prop erty of Hamiltonians for non-conserv ative sys- tems while the second one is sp eciﬁc for the P II equation). 4.2 Linearization and classical-quan tum corresp ondence for P II The linear problems and their compatibility condition for the P II equation ¨ u = 2 u 3 + tu − α are giv en by (2 .1), (2.2) with the mat r ices U =    x 2 + ˙ u − u 2 x − u ( x + u )(2 u 2 − 2 ˙ u + t ) − 2 α − 1 − x 2 − ˙ u + u 2    , V =    x + u 2 1 2 u 2 − ˙ u + t 2 − x + u 2    . (4.5) They are of the f orm implied in section 2.1. Note that u is the simple zero of the right upp er elemen t of the matrix U ( x, t ): b ( u ) = 0. The sp ectral parameter is again delib erately denoted by the same letter x as in the equation of classical motio n (4.1) to stress the fact tha t it is this v ariable ( x -co ordinate of a particle on the line) whic h is going to b e “quantize d” in the “quantum” ve rsion of the P ainlev ´ e-Calogero corresp ondence in the sense that the momentum p = ˙ x is going to b e replaced by the op erator ∂ x . Another meaning of the P II equation (whic h w e will not discuss here) is the condition that the mono drom y data of the ﬁrst linear problem in (2.1) b e indep enden t of the parameter t . 17 It remains t o calculate the p oten tial U ( x, t ) of the no n- stationary Sc hr¨ odinger equa- tion. In t he case of P II ad − bc = − x 4 − tx 2 + (2 α + 1) x − ˙ u 2 + u 4 + tu 2 − (2 α + 1) u , a x − b x A/B = x − u . As a result, w e obta in the no n-stationary Schr¨ odinger equation (in imag inary time) ∂ t ψ = 1 2 ∂ 2 x − x 4 2 − tx 2 2 + αx − t 2 8 − H II ( ˙ u, u ) ! ψ (4.6) or ∂ t ψ =  H II ( ∂ x , x ) − H II ( ˙ u, u )  ψ , (4.7) where H II ( ∂ x , x ) = 1 2 ∂ 2 x − x 4 2 − tx 2 2 + αx − t 2 8 is the quantum Hamiltonian op erator obtained as a literal quantization of the classical Hamiltonian (4.2). The function Ψ( x, t ) = e R t H II ( ˙ u,u ) dt ′ ψ ( x, t ) (4.8) th us ob eys the non-statio na ry Sc hr¨ odinger equation ∂ t Ψ = H II ( ∂ x , x )Ψ =  1 2 ∂ 2 x + V II ( x, t )  Ψ (4.9) without a free t -dep enden t term. T o conclude, w e ha v e tw o equiv alen t represen tations of the P II equation. One is a classical motion in the time-dep enden t p olynomial p otential with Hamiltonian (4.2). The co ordinate of the particle as a function of time obeys the P II equation. Another represen tatio n is a quan tum mec hanical particle in the same time-dep enden t p oten tial. The non-stationary Sc hr¨ odinger equation for this quan tum system in the co ordinate represen tatio n sim ultaneously serv es as the linear problem for time ev olution asso ciated with the Painlev ´ e equation. 5 P ainl e v ´ e IV 5.1 The equ ation The standard form of the P ainlev ´ e IV (P IV ) equation is ∂ 2 t y = ( ∂ t y ) 2 2 y + 3 2 y 3 + 4 ty 2 + 2( t 2 − α ) y + β y , (5.1) where α , β are a rbitrary parameters. This is the ﬁrst example where a change of v ariable is necessary . The time v ariable t remains the same but the dep enden t v ariable should b e c hanged as y = x 2 . This bring s the equation to the Newton form ¨ x = 3 4 x 5 + 2 tx 3 + ( t 2 − α ) x + β 2 x 3 (5.2) 18 whic h admits a Hamiltonian structure similar to the previous examples: ˙ x = ∂ H ( α,β ) IV ∂ p , ˙ p = − ∂ H ( α,β ) IV ∂ x with the classical time-dep enden t Hamilto nian H ( α,β ) IV = H ( α,β ) IV ( p, x ) = p 2 2 − x 6 8 − tx 4 2 − 1 2  t 2 − α  x 2 + β 4 x 2 . (5.3) One may introduce the p oten tial V IV ( x ) = V ( α,β ) IV ( x ) − x 6 8 − tx 4 2 − 1 2  t 2 − α  x 2 + β 4 x 2 , (5.4) then the P IV equation in the Calogero form (5.2) reads ¨ x = − ∂ x V IV ( x ). Note that the partial and f ull time deriv ativ es of the Hamiltonian coincide: ∂ H ( α,β ) IV ∂ t = dH ( α,β ) IV dt = − x 4 ( t ) 2 − tx 2 ( t ) (5.5) (again, the ﬁrst equality is a general prop erty of Hamiltonians for non-conserv ative sys- tems while the second one is sp eciﬁc for the P IV equation). 5.2 Linearization and classical-quan tum c orresp ondence for P IV The system of linear pro blems asso ciated with the P IV equation for the u -v aria ble in the Calogero fo rm, ¨ u = 3 4 u 5 + 2 tu 3 + ( t 2 − α ) u + β 2 u 3 , ( 5 .6) is a mo diﬁed ve rsion of the one giv en in [36]. Their compatibilit y condition is of t he same form (2 .2) with U =        x 3 2 + tx + Q + 1 2 x x 2 − u 2 Q 2 + β 2 u 2 x 2 − Q − α − 1 − x 3 2 − tx − Q + 1 2 x        (5.7) V =        x 2 + u 2 2 + t x − Q + α + 1 x − x 2 + u 2 2 − t        , (5.8) where Q = u ˙ u − u 4 2 − tu 2 . Note that these matrices enjoy the prop ert y b x = 2 B and, t herefore, the no n-stationary Sc hr¨ odinger equation of the form (2 .1 9) is v alid. (Equiv alen tly , w e could start from a rational U – V pair give n in [18]) and tra nsform it to the desired form according to t he 19 strategy o utlined in section 2.4 but in this case the transfor mat ons are simple enough and do not require an y sp ecial consideration.) Note also that u is one of the t w o simple zeros of the right upp er elemen t of the matrix U ( x, t ): b ( u ) = 0. The second zero at t he p oin t x = − u leads to the same results b ecause the equation (5 .6) is inv a rian t under the transformation u → − u . Again, w e delib erately denote the sp ectral parameter b y the same letter x as in the equation of classical motio n (5.2) to stress the fact tha t it is this v ariable ( x -co ordinate of a particle on the line) whic h is going to b e “quantize d” in the “quantum” ve rsion of the P ainlev ´ e-Calogero corresp ondence in the sense that the momentum p = ˙ x is going to b e replaced by the op erator ∂ x . Let us calculate the p oten tial U ( x, t ) in equation (2.19). It consists of t w o parts: one of them is one half of the determinan t of the matrix U and a nother one is − 1 2 a x + A . F or clarit y , w e presen t the results for these tw o parts separately and then tak e the sum. The calculation of the determinan t yields: ad − bc = x 6 4 − tx 4 −  t 2 − α − 1 2  x 2 + β − 1 2 2 x 2 − ˙ u 2 − u 6 4 − tu 4 − ( t 2 − α − 1) u 2 + β 2 u 2 ! − t − Q x 2 . W e see that the v ariables x and u do not completely separate in this expression b ecause of the last term (recall that Q is not a constan t but a dynamical v ar ia ble). F ortunately , this term cancels out after adding the second part of the p oten tial: − a x + 2 A = − x 2 2 + 1 2 x 2 + u 2 + t + Q x 2 . Com bining the t w o parts together, w e get: 1 2 ( ad − bc − a x + 2 A ) = x 6 8 − tx 4 2 − 1 2  t 2 − α  x 2 + β + 1 2 4 x 2 − ˙ u 2 2 − u 6 8 − tu 4 2 − 1 2 ( t 2 − α ) u 2 + β 4 u 2 ! . (5.9) Therefore, equation (2.19) reads ∂ t ψ =  H ( α,β + 1 2 ) IV ( ∂ x , x ) − H ( α,β ) IV ( ˙ u, u )  ψ , (5.10) where H ( α,β + 1 2 ) IV ( ∂ x , x ) = 1 2 ∂ 2 x − x 6 8 − tx 4 2 − 1 2 ( t 2 − α ) x 2 + β + 1 2 4 x 2 . The function Ψ( x, t ) = e R t H ( α,β ) IV ( ˙ u,u ) dt ′ ψ ( x, t ) (5.11) th us ob eys the non-statio na ry Sc hr¨ odinger equation ∂ t Ψ = H ( α,β + 1 2 ) IV ( ∂ x , x )Ψ =  1 2 ∂ 2 x + V ( α,β + 1 2 ) IV ( x, t )  Ψ (5.12) 20 without a fr ee t -dep enden t term. Note the shift β → β + 1 2 whic h can b e thought of as a “quantum correction”. T o conclude, we ha v e t w o equiv alen t represen t a tions o f the P IV equation. One is a classical motion in the t ime-dep enden t p otential with Hamiltonian (5.3). The co ordinate of the particle as a function of time ob eys the P IV equation. Another represen tatio n is a quan tum mec hanical particle in the time-dep enden t p oten tial of the same form, with the mo diﬁed co eﬃcien t in front of 1 /x 2 . The non-stationary Sch r¨ odinger equation for this quan tum system in the co ordinate represen tation simultaneously serv es as the linear problem f or time ev olution asso ciated with the P ainlev ´ e equation. 6 P ainl e v ´ e I I I 6.1 The equ ation The standard form of the P II I equation for a function y ( T ) is ∂ 2 T y = ( ∂ T y ) 2 y − ∂ T y T + 1 T ( αy 2 + β ) + γ y 3 + δ y , (6.1) where α , β , γ , δ are arbitrar y parameters. The ch ange o f the v ariables T = e t , y = e 2 x brings this equation to the Newton form 2 ¨ x = e t ( αe 2 x + β e − 2 x ) + e 2 t ( γ e 4 x + δ e − 4 x ) . (6.2) Note that only tw o parameters of the f o ur are really indep endent b ecause the other t w o can b e eliminated by the shifts x → x 0 , t → t 0 with constan t x 0 , t 0 . How ev er, w e will k eep 3 parameters in or der to b e able t o consider some particular cases whic h are not reac hable otherwise. So, equation (6.2) acquires the form ¨ x = 2 ν 2 e t sinh(2 x − 2  ) + 4 µ 2 e 2 t sinh(4 x ) , (6.3) where ν , µ,  are the parameters. In principle, one of them, say ν can b e put equal t o one b y the shift t → t − 2 lo g ν but this w orks only if ν 6 = 0. Equation (6.3) admits a Hamiltonia n structure similar to the previous examples: ˙ x = ∂ H II I ∂ p , ˙ p = − ∂ H II I ∂ x with the classical time-dep enden t Hamilto nian H II I = H II I ( p, x ) = p 2 2 − ν 2 e t cosh(2 x − 2  ) − µ 2 e 2 t cosh(4 x ) . (6.4) One may introduce the p oten tial V II I ( x ) = − ν 2 e t cosh(2 x − 2  ) − µ 2 e 2 t cosh(4 x ) , (6.5) then the P II I equation reads ¨ x = − ∂ x V II I ( x ). 21 The case µ = 0 is sp ecial. In this case, one can put  = 0 without loss of generality , so the P II I equation acquires the form ¨ x = 2 ν 2 e t sinh(2 x ) (6.6) with j ust one parameter ν (whic h in fact can b e eliminated b y a shift of time) and with the classical Hamilto nian H II I = p 2 2 − ν 2 e t cosh(2 x ) . (6.7) This equation will b e referred to as truncated P II I equation. 6.2 The U – V pairs for P II I 6.2.1 The case of the t runcated P II I equation The truncated P II I equation ¨ u = 2 ν 2 e t sinh(2 u ) (6.8) is the compatibility condition for linear problems with matrices U , V of rather simple form. Indeed, it is easy to c hec k tha t the zero curv a ture condition (2.2) with U ( x, t ) =    ˙ u 2 ν e t/ 2 sinh( x − u ) 2 ν e t/ 2 sinh( x + u ) − ˙ u    , (6.9) V ( x, t ) =    0 ν e t/ 2 cosh( x − u ) ν e t/ 2 cosh( x + u ) 0    (6.10) yields equation (6.8) 2 . Moreo v er, these matrices o bviously satisfy the condition b x = 2 B and u is the ﬁrst order zero of the elemen t b ( x ) (of course t here are inﬁnitely many zeros in the complex x -pla ne a t the p oin ts of the lattice u + π i Z but all of them ob ey the same equation (6.8)). 6.2.2 The general case In the general case the U – V pair for the P II I equation is more complicated. W e take the linear problems for P II I giv en in [18] as a starting p oin t, passing to the exp onen tial parametrization from the ve ry b eginning and then transform t hem to the ones a ppropriate for o ur purp ose. So, w e start with the linear problems ∂ x ˜ Ψ =    e 2 x + t − g 11 e − 2 x + t + θ + 1 2 2 v e x − t 2 − g 12 e − x + t 2 2 w e − x + t 2 − g 21 e − 3 x + 3 t 2 − e 2 x + t + g 11 e − 2 x + t − θ − 1 2    ˜ Ψ (6.11) 2 The Lax pa ir s for P II I and P V were obtained by G.Amino v and S.Arthamonov v ia trigonometr ic scaling limits from the one found in [40]. How ever, the condition b x = 2 B for the Lax pa irs obtained in this way holds in the case of the truncuted P II I equation only and do es not hold in gene r al. 22 ∂ t ˜ Ψ = 1 2    e 2 x + t + g 11 e − 2 x + t − 1 2 2 v e x − t 2 + g 12 e − x + t 2 2 w e − x + t 2 + g 21 e − 3 x + 3 t 2 − e 2 x + t − g 11 e − 2 x + t + 1 2    ˜ Ψ (6.12) where v , w , g 11 , g 12 , g 21 are y et unkno wn functions of t and θ is a parameter. The functions g ik are naturally though t of as entrie s of a traceless matrix G = g 11 g 12 g 21 − g 11 ! . (6.13) Note that the x -deriv ativ e of the righ t upp er elemen t of the ˜ U -matrix in (6.11) is just equal to twice the right upp er elemen t o f the ˜ V -mat r ix in (6.12). As b efore, w e delib erately denote t he sp ectral parameter b y the same letter x as in the equation of t he classic al motion (6.2) to stress the fact that it is this v ariable ( x - co ordinate of a particle on the line) whic h is going to b e “quan tized” in the “quan tum” v ersion of the P ainlev ´ e-Calogero corresp ondence in the sense that the momen tum p = ˙ x is g o ing to b e replaced by the op erator ∂ x . The compatibility of the linear problems (6 .11),(6.12) implies the following system of diﬀeren tial equations:                                  ˙ g 11 = 2( v g 21 − w g 12 ) ˙ g 12 = θ g 12 − 4 v g 11 ˙ g 21 = − θ g 21 + 4 w g 11 ˙ v = − θ v − g 12 e 2 t ˙ w = θ w + g 21 e 2 t . (6.14) Com bining these equations, one easily ﬁnds t w o in tegrals: χ := g 2 11 + g 12 g 21 (6.15) λ := v g 21 + w g 12 + θ g 11 , (6.16 ) where χ and λ a re inte gratio n constan ts. Note that t he ﬁrst integral is just determinan t of the matrix G (6.13) with opp osite sign. Using (6.15), ( 6 .16), one can exclude w a nd g 21 , w = λ − θ g 11 − v g 21 g 12 , g 21 = χ − g 2 11 g 12 , and reduce the system (6.14) to a simpler o ne:                ˙ g 11 = 4 v g − 1 12 ( χ − g 2 11 ) + 2 θ g 11 − 2 λ ˙ g 12 = θ g 12 − 4 v g 11 ˙ v = − θ v − g 12 e 2 t . (6.17) 23 F urther, these equations imply the following system f o r the functions f = v / g 12 , g = g 11 :      ˙ f = 4 g f 2 − 2 θ f − e 2 t ˙ g = − 4 f g 2 + 2 θ g + 4 χf − 2 λ . (6.18) No w, substituting g = ˙ f + 2 θ f + e 2 t 4 f 2 from the ﬁrst equation into the second one, we obtain a closed equation for f , ¨ f = ˙ f 2 f + 16 χf 3 − 8 λf 2 − 2( θ + 1) e 2 t − e 4 t f (6.19) whic h is equiv alen t to the P II I equation (6.1) and can b e brought to the origina l form by the c hange of v ariable T = e t . The ch ange of the dep enden t v aria ble f = e − 2 u + t yields the equation ¨ u = e t  ( θ + 1) e 2 u + 4 λe − 2 u  + 1 2 e 2 t  e 4 u − 16 χe − 4 u  (6.20) whic h has the form (6.3 ) with µ = 1 / 2 under the iden tiﬁcation of parameters θ + 1 = ν 2 e − 2  , 4 λ = − ν 2 e 2  , χ = 1 16 . 6.3 Classical-quan tum corresp ondence for P II I 6.3.1 The case of truncated P II I equation Let us start with the simplest case µ = 0 and use the U – V pair (6.9 ) , (6 .10). A simple calculation sho ws that in this case the linear equation for ψ (2.17) b ecomes the “non- stationary Mathieu equation” ∂ t ψ =  H II I ( ∂ x , x ) − H II I ( ˙ u, u )  ψ , (6.21) where H II I ( ∂ x , x ) = 1 2 ∂ 2 x − ν 2 e t cosh(2 x ) , i.e., w e again observ e a p erfect classical-quan tum corresp ondence. Note t ha t in this case a x = A = 0 , so the p oten tial is giv en solely by determinant of the matrix U : 1 2 ( ad − bc ) = − ˙ u 2 2 − 2 ν 2 e t sinh( x + u ) sinh( x − u ) = − ν 2 e t cosh(2 x ) − ˙ u 2 2 − ν 2 e t cosh(2 u ) ! . W e remark that the non-stationar y Mathieu equation in connection with the P II I equation w as men tioned in [42]. 24 6.3.2 The general case In order to ac hiev e a precise classical-quan t um corresp ondence in the general case of the P II I equation with arbitrary parameters, o ne should mo dify the syste m of linear problems giv en a bov e b y a diagonal x -indep enden t (but t -dep enden t) gaug e transformation o f the form (2 .9) with ω = g − 1 2 12 (2 f e − t ) − 1 4 , (6.22) where f = v /g 12 as b efore. Another small mo diﬁcation whic h is necessary to a chiev e p erfect classical-quan tum corresp ondence is the shift of the sp ectral parameter x → x − 1 2 log 2. Then the linear problems (6.11), (6.12) acquire the form (2.1) with U =     1 2 e 2 x + t − 2 g 11 e − 2 x + t + θ + 1 2 f 1 2 e x − f − 1 2 e − x + t 4( v g 12 ) 1 2  w e − x − g 21 e − 3 x + t  − 1 2 e 2 x + t + 2 g 11 e − 2 x + t − θ − 1 2     (6.23) V =     1 4 e 2 x + t + g 11 e − 2 x + t + h 1 2  f 1 2 e x + f − 1 2 e − x + t  2( v g 12 ) 1 2  w e − x + g 21 e − 3 x + t  − 1 4 e 2 x + t − g 11 e − 2 x + t − h     , (6.24) where h := ∂ t log  f − 1 4 g − 1 2 12  = ˙ f 4 f + e 2 t 2 f + θ 2 . (6.25) Recall also that f = e − 2 u + t , so the right upp er elemen t of the ma t r ix U is b ( x ) = 2 e t/ 2 sinh( x − u ) and u is its ﬁrst order zero. No w, the calculation of the p oten tial U ( x, t ) in the non-stationary Schr¨ odinger equa- tion (2.19) yields U ( x, t ) = − e 2 t 8  e 4 x + 16 χe − 4 x  − e t 2  ( θ + 1) e 2 x − 4 λe − 2 x  − ˙ u 2 2 + e 2 t 8  e 4 u + 16 χe − 4 u  + e t 2  ( θ + 1) e 2 u − 4 λe − 2 u  , (6.26) so the Schr¨ odinger equation a cquires the desired form ∂ t ψ =  H II I ( ∂ x , x ) − H II I ( ˙ u, u )  ψ , (6.27) where H II I ( ∂ x , x ) = 1 2 ∂ 2 x − e 2 t 8  e 4 x + 16 χe − 4 x  − e t 2  ( θ + 1) e 2 x − 4 λe − 2 x  . The function Ψ( x, t ) = e R t H II I ( ˙ u,u ) dt ′ ψ ( x, t ) (6.28) 25 th us ob eys the non-statio na ry Sc hr¨ odinger equation ∂ t Ψ = H II I ( ∂ x , x )Ψ =  1 2 ∂ 2 x + V II I ( x, t )  Ψ (6.29) with the same classical p otential (6.5). T o conclude, we ha v e tw o equiv alen t represen tations of t he P II I equation. One is a classical motion in the t ime-dep enden t p otential with Hamiltonian (6.4). The co ordinate of t he particle as a function of time ob eys the P II I equation. Another represen t ation is a quan tum mec hanical par ticle in the same time-dep enden t p oten tial describ ed by a non- stationary Sc hr¨ odinger equation. The latter sim ultaneously serv es as the linear problem for t ime ev olution asso ciated with the Painle v ´ e equation. 7 P ainl e v ´ e V 7.1 The equ ation The standard form of the P V equation is ∂ 2 T y = 1 2 y + 1 y − 1 ! ( ∂ T y ) 2 − ∂ T y T + y ( y − 1) 2 T 2 α + β y 2 + γ T ( y − 1) 2 + δ T 2 ( y + 1) ( y − 1) 3 ! , (7.1) where α , β , γ , δ are ar bitr ary pa rameters. A re-scaling o f the dep enden t v ariable allows one to ﬁx one of these pa r ameters, so there are three essen tially indep enden t para meters . The change of the time v ariable T = e 2 t allo ws o ne to eliminate the term ∂ T y /T in the righ t hand side, so that the equation b ecomes ¨ y = 1 2 y + 1 y − 1 ! ˙ y 2 + 4( y − 1) 2  αy + β y  + 4 γ e 2 t y + 4 δ e 4 t y ( y + 1) y − 1 . (7.2) F urther, the c hange of the dep enden t v ariable y = coth 2 x (7.3) brings the equation to the for m ¨ x = − 2 α cosh x sinh 3 x − 2 β sinh x cosh 3 x − γ e 2 t sinh(2 x ) − 1 2 δ e 4 t sinh(4 x ) (7.4) whic h can b e written as the Newton equation ¨ x = − ∂ x V V ( x ) (7.5) with the time-dep enden t p otential V V ( x ) = − α sinh 2 x − β cosh 2 x + γ e 2 t 2 cosh(2 x ) + δ e 4 t 8 cosh(4 x ) . (7.6) 26 Again, we see that only three parameters among the fo ur are really indep enden t b ecause one of t hem can b e put equal to 1 b y a prop er shift of t . This equation admits a Hamiltonian structure similar to the previous cases: ˙ x = ∂ H V ∂ p , ˙ p = − ∂ H V ∂ x with the classical time-dep enden t Hamilto nian H V ( p, x ) = p 2 2 + V V ( x ) (7.7) T o indicate the dep endence on the parameters, w e will write H V ( p, x ) = H ( α,β ,γ ,δ ) V ( p, x ). 7.2 The z ero curv ature represen tation of the P V equation The choice of the U – V pair for the P V equation suitable for our purp ose is by no means ob vious. W e start from a mo diﬁed v ersion of the U – V pair with rational dep endence on the sp ectral parameter suggested by M.Jim b o a nd T.Miw a [18] and then show ho w to transform it to the desired form. 7.2.1 The mo diﬁed Jimbo-Miw a U – V p air for P V Let us consider the system of linear pr o blems      ∂ X Ψ = U ( X , t ) Ψ ∂ t Ψ = V ( X , t ) Ψ (7.8) with the matrices U =        e 2 t 2 + g X − g + σ X − 1 v X − w X − 1 v 1 X − w 1 X − 1 − e 2 t 2 − g X + g + σ X − 1        (7.9) V =    X e 2 t 2( v − w ) 2( v 1 − w 1 ) − X e 2 t    (7.10) and a column 2-comp onen t v ector Ψ . Here X is the sp ectral parameter, v , w , v 1 , w 1 , g are some functions of t to b e constrained b y the zero curv ature condition ∂ X V − ∂ t U + [ V , U ] = 0 and σ is an arbitrary constan t. The zero curv ature condition yields t he system o f 27 diﬀeren tial equations                                  ˙ g = 2 ( v w 1 − w v 1 ) ˙ v = − 4( v − w ) g ˙ w = − 4( v − w )( g + σ ) + 2 w e 2 t ˙ v 1 = 4( v 1 − w 1 ) g ˙ w 1 = 4( v 1 − w 1 )( g + σ ) − 2 w 1 e 2 t . (7.11) Com bining these equations, one easily ﬁnds t w o in tegrals: v v 1 + g 2 = ζ 2 , w w 1 + g ( g + 2 σ ) = ξ 2 + 2 ξ σ , (7.12) where ζ , ξ are arbitrary constants (the in tegration constants are expresse d in this par- ticular w a y for later conv enience). These form ulas a llow one to substitute v 1 = ( ζ − g )( ζ + g ) v , w 1 = ( ξ − g )(2 σ + ξ + g ) w in to t he ﬁrst equation of the system (7 .1 1) thus reducing it to the system of three equa- tions for three unkno wn functions. Let us introduce t he function y = v w , (7.13) then the ﬁrst equation of the system (7.11) b ecomes ˙ g = 2  y − 1 ( g + ξ )( g − ξ ) − y ( g + ξ + 2 σ ) ( g − ξ )  . (7.14) W riting ˙ y = ˙ v w − v ˙ w w 2 = ˙ v w − y ˙ w w , w e ﬁnd from the second a nd third equations of the system (7.11): ˙ y = 4( y − 1) 2 g + 4 σ y ( y − 1) − 2 y e 2 t . (7.15) Plugging g = ˙ y + 2 y e 2 t 4( y − 1) 2 − σ y y − 1 (7.16) expresse d from this equation in terms of y in to (7 .14), o ne obtains, after a relativ ely lo ng calculation, ¨ y =  1 2 y + 1 y − 1  ˙ y 2 + 8( y − 1) 2  ( ξ + σ ) 2 y − ζ 2 y  + 4(2 σ − 1 ) e 2 t y − 2 e 4 t y ( y + 1) y − 1 (7.17) whic h is the P V equation in the for m (7 .2) with α = 2( ξ + σ ) 2 , β = − 2 ζ 2 , γ = 2 σ − 1 , δ = − 1 2 . (7.18) 28 7.2.2 Hyp erb olic parametrization The crucial step of the further construction is a parametrizatio n o f the mo diﬁed Jim b o- Miw a U – V pair (7.9), (7.1 0 ) in terms of hy p erb olic functions. This parametrization corresp onds to the hyperb olic substitution (7.3) fo r the dep enden t v aria ble leading to the Calogero form of the P V equation but do es not coincide with it. The next step is a sp ecial diago nal gauge transformatio n whic h recasts the matrices in the form suc h that the conditio n b x = 2 B (2.4) is satisﬁed. The required h yp erbo lic parametrization is ac hiev ed by setting X = cosh 2 x. (7.19) Since this tr ansformation do es not dep end on t , the general form ulas (2.27 ) simplify . T aking in to accoun t the rule ∂ x = 2 cosh x sinh x ∂ X b y whic h the deriv ativ e ∂ X is transformed, w e see that the ﬁrst linear problem in (7.8) should b e c hanged to ∂ x Ψ = 2 cosh x sinh x U ( X ( x ) , t ) Ψ , so the U – V pair in the h yp erb olic pa r a metrization acquires the f orm ˜ U ( x, t ) =    ˜ a 2 v tanh x − 2 w coth x 2 v 1 tanh x − 2 w 1 coth x − ˜ a    (7.20) with ˜ a = e 2 t sinh x cosh x + 2 g tanh x −  2 g + 2 σ  coth x and ˜ V ( x, t ) =    e 2 t cosh 2 x 2( v − w ) 2( v 1 − w 1 ) − e 2 t cosh 2 x    . (7.21) Here the functions v , w , v 1 , w 1 , g are the same as in (7.9 ) , (7.10 ). Clearly , the zero cur- v ature condition yields equation (7.17) with the same constants ζ , ξ as in (7.1 2 ). This U – V pair ob eys the prop erty b x = 2 B . W e delib erately denote the h yp erb o lic sp ectral parameter by the same letter x as in the equation of t he classic al motion (7.4) to stress the fact that it is this v ariable ( x - co ordinate of a particle on the line) whic h is going to b e “quan tized” in the “quan tum” v ersion of the P ainlev ´ e-Calogero corresp ondence in the sense that the momen tum p = ˙ x is g o ing to b e replaced by the op erator ∂ x . 7.3 Classical-quan tum corresp ondence for P V In order to a c hiev e the precise classical-quan tum cor r espondence, one should apply a diagonal gauge transformation. Namely , let us pass to the gauge equiv alent U – V pair U = Ω − 1 ˜ U Ω − Ω − 1 ∂ x Ω , V = Ω − 1 ˜ V Ω − Ω − 1 ∂ t Ω with Ω =          e − t ( v − w ) sinh x cosh x ! 1 2 0 0 e − t ( v − w ) sinh x cosh x ! − 1 2          . 29 Explicitly , the U – V pair (7.2 0), (7.21) t r a nsforms into U ( x, t ) =        a 2 e t ( v sinh 2 x − w cosh 2 x ) v − w 2 e − t ( v − w )  v 1 cosh 2 x − w 1 sinh 2 x  − a        (7.22) with a = e 2 t sinh x cosh x +  2 g + 1 2  tanh x −  2 g + 2 σ − 1 2  coth x and V ( x, t ) =       e 2 t  cosh 2 x + sinh 2 u  − 2 σ + 1 2 e t sinh(2 x ) 4( v − w )( v 1 − w 1 ) e t sinh(2 x ) − e 2 t  cosh 2 x + sinh 2 u  + 2 σ − 1 2       . (7.23) In pr inciple, the auxiliary functions g , v , w , v 1 , w 1 can b e excluded from the h yp erbo lic U – V pair (7 .22), (7.23), with the ﬁnal result b eing written solely in t erms of u and ˙ u . Ho w ev er, for the purp ose of this pap er w e do not need this form (it will b e presen ted elsewhere ). The result of the previous subsections imply that setting y = v w = coth 2 u w e ﬁnd from the zero curv ature condition for the h yp erb olic U – V pair (7.22), (7 .23) t he P V equation in the Calogero- Inozem tsev-lik e form: ¨ u = − 4( ξ + σ ) 2 cosh u sinh 3 u + 4 ζ 2 sinh u cosh 3 u − (2 σ − 1) e 2 t sinh(2 u ) + 1 4 e 4 t sinh(4 u ) . (7.24) Note that in this parametrization b ( x ) = 2 e t sinh( x − u ) sinh ( x + u ). F or real u this elemen t has just tw o zeros in the strip | Im x | < π at the p oin ts ± u and t he b oth ob ey the same equation (7 .24). In order to c hec k the classical-quan tum corresp o ndenc e, one should calculate the p oten tial U ( x, t ) of the non-stationary Sc hr¨ odinger equation (2.19). The calculation is straigh tforward and the result is U ( x, t ) = 4 ζ 2 − 1 4 2 cosh 2 x − 4( ξ + σ ) 2 − 1 4 2 sinh 2 x − e 4 t 16 cosh(4 x ) +  σ − 1 2  e 2 t cosh(2 x ) − ˜ H , (7.25) where ˜ H = 2( v − w )( v 1 − w 1 ) − e 4 t 16 − e 2 t  2 g + w v − w + σ + 1 2  + 2 σ 2 . (7.26) The x -dep enden t part of the p oten tia l coincides with the p otential (7.4) up to some shifts of the par a meters α → α − 1 8 , β → β + 1 8 (see (7.18)). Let us ﬁnd the x -indep enden t term ˜ H and compare it with the classical Hamiltonian fo r P V . Using (7.12), w e g et: 2( v − w )( v 1 − w 1 ) = 2( y − 1)  y − 1 y g 2 + 2 σ g + ζ 2 y − ξ 2 − 2 ξ σ  30 and g is giv en b y (7.16). Pass ing to the h yp erb olic parametrization, w e ha v e: g = − ˙ u 2 sinh u cosh u + e 2 t 2 sinh 2 u cosh 2 u − σ cosh 2 u and 2( v − w )( v 1 − w 1 ) = ˙ u 2 2 + e 4 t 16 (cosh(4 u ) − 1) − e 2 t 2 sinh(2 u ) ˙ u + 2 ζ 2 cosh 2 u − 2( ξ + σ ) 2 sinh 2 u − 2 σ 2 . Plugging a ll this in to (7.26), w e get exactly the classical Ha milto nian H ( α,β ,γ ,δ ) V ( ˙ u, u ) with the pa rameters α , β , γ , δ give n by (7.1 8): ˜ H = ˙ u 2 2 − 2( ξ + σ ) 2 sinh 2 u + 2 ζ 2 cosh 2 u + e 2 t 2 (2 σ − 1) cosh(2 u ) − e 4 t 16 cosh(4 u ) = H ( α,β ,γ ,δ ) V ( ˙ u, u ) . (7.27) Summing up, in the case of P V w e ha v e the non-stationa r y Sc hr¨ odinger equation ∂ t ψ =  H ( α − 1 8 , β + 1 8 , γ , 1 2 ) V ( ∂ x , x ) − H ( α,β ,γ , 1 2 ) V ( ˙ u, u )  ψ . (7.28) The function Ψ( x, t ) = e R t H ( α, β , γ , 1 2 ) V ( ˙ u,u ) dt ′ ψ ( x, t ) (7.29) th us ob eys the non-statio na ry Sc hr¨ odinger equation ∂ t Ψ = H ( α − 1 8 , β + 1 8 , γ , 1 2 ) V ( ∂ x , x )Ψ =  1 2 ∂ 2 x + V ( α − 1 8 , β + 1 8 , γ , 1 2 ) V ( x, t )  Ψ . (7.30) Note that the parameters α , β in the quan tum Hamiltonian are shifted b y ± 1 8 . T o conclude, w e hav e t wo equiv alen t represen tations of the P V equation. O ne is a classical motion in the t ime-dep enden t p otential with Hamiltonian (7.7). The co ordinate of the particle as a function o f time ob eys the P V equation. Another represen ta t ion is a quan tum mec hanical particle in the time-dep enden t p oten tial of the same f o rm with mo diﬁed co eﬃcien ts describ ed by a non-statio nary Sc hr¨ odinger equation. The latt er si- m ultaneously serv es a s the linear problem for time ev olution asso ciated with the P ainlev ´ e equation. 8 P ainl e v ´ e VI 8.1 The equ ation The standard form of the P VI equation is ∂ 2 T y = 1 2 1 y + 1 y − 1 + 1 y − T ! ( ∂ T y ) 2 − 1 T + 1 T − 1 + 1 y − T ! ∂ T y + y ( y − 1)( y − T ) T 2 ( T − 1) 2 α + β T y 2 + γ ( T − 1) ( y − 1) 2 + δ T ( T − 1 ) ( y − T ) 2 ! , (8.1) 31 where α , β , γ , δ are arbitrary par a meters . Let us p erform the c hange of the v ariables ( y , T ) → ( x, t ) g iven by the form ulas [28, 27] y = ℘ ( x ) − e 1 e 2 − e 1 , T = e 3 − e 1 e 2 − e 1 , (8.2) where ℘ ( x ) = ℘ ( x | 1 , τ ) is the W eierstrass ℘ -function with p erio ds 1, τ = 2 π it , and e k = ℘ ( ω k ), k = 1 , 2 , 3, are the v alues of ℘ ( x ) at t he half- perio ds ω 1 = 1 2 , ω 2 = 1 2 (1 + τ ), ω 3 = 1 2 τ . It is con v enien t to set a lso ω 0 = 0. This c hange of v ariables br ing s the P VI equation to the Newton form ¨ x = 3 X k =0 ν k ℘ ′ ( x + ω k ) , (8.3) where ℘ ′ ( x ) ≡ ∂ x ℘ ( x ), and ν 0 = α , ν 1 = − β , ν 2 = γ , ν 3 = − δ + 1 2 . This equation a dmits the Ha miltonian structure ˙ x = ∂ H VI ∂ p , ˙ p = − ∂ H VI ∂ x with the classical time-dep enden t Hamilto nian H VI ( p, x ) = p 2 2 + V VI ( x ) , V VI ( x ) = − 3 X k =0 ν k ℘ ( x + ω k ) . (8.4) It describ es classical motio n of a p oin t-lik e particle in the p erio dic time-dep enden t po- ten tial. The time dep endence is hidden in the second p erio d of the ℘ -function: ℘ ( x ) = ℘ ( x | 1 , τ ) , τ = 2 π it. (8.5) T o indicate the dep endenc e on the parameters, w e will write H VI ( p, x ) = H ( α,β ,γ ,δ ) VI ( p, x ) and V VI ( x, t ) = V ( α,β ,γ ,δ ) VI ( x, t ). The elliptic fo rm of the P VI equation was discussed also in [37, 38 ]. 8.2 The z ero curv ature represen tation of the P VI equation Diﬀeren t v ersions of the U – V pairs for the P VI equation with sp ectral parameter on an elliptic curv e w ere found in [39] for the special case of equal constan ts ν k = ν and in [40] for the general case. How ev er, they app ear to b e unsuitable for o ur purp ose. Like in the case of the P V equation, w e start from a mo diﬁed v ersion of the U – V pair with rational dep endence o n the sp ectral parameter suggested in [18], then pass to an elliptic parametrization and transform it to the desired form by a ga uge transformatio n. 8.2.1 The mo diﬁed Jimbo-Miw a U – V p air for P VI Let us consider the system of linear pr o blems      ∂ X Ψ = U ( X , T ) Ψ ∂ T Ψ = V ( X , T ) Ψ (8.6) 32 with the matrices [18 ] U =        g 0 + ξ 0 X + g 1 + ξ 1 X − 1 + g 2 + ξ 2 X − T −  u 0 g 0 X + u 1 g 1 X − 1 + u 2 g 2 X − T  g 0 + 2 ξ 0 u 0 X + g 1 + 2 ξ 1 u 1 ( X − 1) + g 2 + 2 ξ 2 u 2 ( X − T ) − g 0 + ξ 0 X + g 1 + ξ 1 X − 1 + g 2 + ξ 2 X − T !        (8.7) V =        − g 2 + ξ 2 X − T u 2 g 2 X − T − g 2 + 2 ξ 2 u 2 ( X − T ) g 2 + ξ 2 X − T        (8.8) and t w o-comp onen t v ector Ψ . Here X is the spectral parameter living o n the Rie- mann sphere, g i , u i are some functions of T to b e determined from the zero curv ature condition ∂ X V − ∂ T U + [ V , U ] = 0 and ξ i are arbitrary constan ts. Belo w in this se c- tion w e denote the en tries of the matrices U , V as U = U ( X ) = a ( X ) b ( X ) c ( X ) d ( X ) ! , V = V ( X ) = A ( X ) B ( X ) C ( X ) D ( X ) ! (for brevit y , the T -dep endence is not indicated explic- itly). The following in tegrals o f motion a re immediate consequences of the zero curv ature condition: g 0 + g 1 + g 2 = ξ 3 u 0 g 0 + u 1 g 1 + u 2 g 2 = 0 g 0 + 2 ξ 0 u 0 + g 1 + 2 ξ 1 u 1 + g 2 + 2 ξ 2 u 2 = 0 . (8.9) Here ξ 3 is a n arbitrar y constan t, the v alues of the o ther t w o in tegrals are set equal to zero following [18]. The full system of ordinary diﬀeren tial equations for the functions g i , u i whic h follow s from the zero curv ature condition is explicitly giv en in App endix A. Next, let us in tro duce a function y b y represen ting the right upp er en try of t he matrix U in the form b ( X ) = K ( X − y ) X ( X − 1)( X − T ) , (8.10) where K = T u 0 g 0 + ( T − 1) u 1 g 1 , y = T u 0 g 0 K . (8.11) Note that in terms of K, y w e ha v e: u 0 g 0 = K y T , u 1 g 1 = − K ( y − 1) T − 1 , u 2 g 2 = K ( y − T ) T ( T − 1) . (8 .12) One can see that the zero curv ature condition implies the P VI equation (8.1) fo r the function y with α = 2  ξ + 1 2  2 , β = − 2 ξ 2 0 , γ = 2 ξ 2 1 , δ = 1 2 − 2 ξ 2 2 , (8.13 ) 33 where ξ = ξ 0 + ξ 1 + ξ 2 + ξ 3 . (8.14) Some details of t he pro of are presen ted in App endix A. 8.2.2 Elliptic parametrization The crucial ingredien t of the construction is a parametrization of the mo diﬁed Jimbo- Miw a U – V pair (8.7), (8.8) in terms of elliptic functions. This pa r a metrization corre- sp onds to the elliptic substitution (8.2 ) for the dep enden t and indep enden t v ariables leading to the Calogero fo r m o f the P VI equation. W e use the general relations fo r a c hange of v ariables from X , T to x, t of the form X = X ( x, t ), T = T ( t ) giv en in section 2.4. According to these relat io ns, the U – V pair in the v ariables x, t is ˜ U ( x, t ) = ∂ X ∂ x U ( X ( x, t ) , T ( t )) ˜ V ( x, t ) = ∂ T ∂ t V ( X ( x, t ) , T ( t )) + ∂ X ∂ t U ( X ( x, t ) , T ( t )) , (8.15) where the en tries of the matrices U , V in t he right hand side should b e expressed t hr o ugh the new v ariables x, t (see (8.15)). W e need some formu las whic h would allo w us to mak e this transformation explicit. It is na tural to exp ect that the c hange of the time v ariable is the same as for the P VI equation itself (see the second for mula in (8 .2 )). It turns out that the c ha ng e of the sp ectral pa rameter is also giv en b y the same elliptic function as the one used for the dep enden t v ariable in (8.2): X ( x, t ) = ℘ ( x ) − e 1 e 2 − e 1 , T ( t ) = e 3 − e 1 e 2 − e 1 = ϑ 3 (0 | τ ) ϑ 0 (0 | τ ) ! 4 . (8.16) Here ℘ ( x ) = ℘ ( x | 1 , τ ) and e j = ℘ ( ω j | 1 , τ ) dep end on the new time v ariable t = τ 2 π i through the second p erio d of the ℘ -function. In the last form ula, w e give the par a metriza- tion of T in terms of Jacobi’s theta- functions ϑ a ( x | τ ) (see App endix B). The argumen ts leading to equations (8 .16) a nd the deriv ation are giv en in App endix C. L et us also no te that the elliptic substitution for the sp ectral parameter of the form (8.16) w as ﬁrst sug- gested in [41], where the relation b et w een rational and elliptic f orms of the linear problems for the P VI equation w as described in terms o f mo diﬁcation o f the cor r esp onding vec tor bundles. Similar to the previously considered cases, we delib erately denote the elliptic sp ectral parameter b y the same letter x as in the equation of the classical motion (8.3 ) to stress the fa ct that it is this v ariable ( x - co ordinat e of a pa rticle on the line) whic h is going to b e “quan tized” in the “quantum” v ersion of the Painlev ´ e-Calogero corresp ondence in the sense that t he momentum p = ˙ x is g oing to b e replaced by the op erator ∂ x . F or practical calculations w e need some mo r e fo r mulas. First of all, w e ha v e X = ℘ ( x ) − e 1 e 2 − e 1 , X − 1 = ℘ ( x ) − e 2 e 2 − e 1 , X − T = ℘ ( x ) − e 3 e 2 − e 1 , (8.17) 34 so that the iden tities ∂ X ∂ x ! 2 = 4( e 2 − e 1 ) X ( X − 1) ( X − T ) (8.18) ∂ 2 X ∂ x 2 = 2( e 2 − e 1 ) X ( X − 1) ( X − T )  1 X + 1 X − 1 + 1 X − T  (8.19) hold true. (The ﬁrst one is the diﬀerential equation for the ℘ - function (B13), the second one is a result of its further diﬀeren tiating with resp ect to x .) Next w e need the fo llo wing relations: ( e 2 − e 1 ) T X = ℘ ( x + ω 1 ) − e 1 − ( e 2 − e 1 )( T − 1) X − 1 = ℘ ( x + ω 2 ) − e 2 ( e 2 − e 1 ) T ( T − 1) X − T = ℘ ( x + ω 3 ) − e 3 . (8.20) A t last, let us presen t form ulas fo r deriv ativ es of the elliptic functions with resp ect t o t = τ 2 π i . All of them follow from the “heat equation” ob ey ed by an y Jacobi’s theta- function ϑ a ( x ) = ϑ a ( x | τ ), a = 0 , . . . , 3: 2 ∂ t ϑ a ( x ) = ∂ 2 x ϑ a ( x ) , t = τ 2 π i . (8.21) In pa rticular, we need the f o llo wing tw o deriv atives : ∂ X ∂ t = ∂ X ∂ x ϑ ′ 0 ( x ) ϑ 0 ( x ) (8.22) ∂ T ∂ t = 2( e 2 − e 1 ) T ( T − 1) . (8.23) The deriv ation is giv en in App endix B. The formu la f or ∂ X/∂ t ﬁrst a pp eared in T ak asaki’s pap er [29 ]. Note that diﬀeren tia ting a double-p erio dic function of x with resp ect to one of the p erio ds, as in (8.22), we obtain a function which is not an elliptic f unction of x . The second formula is a dir ect coro lla ry of the deﬁnition and (B14). (In fact, since T = X ( ω 3 ), the second formula follows from the ﬁrst o ne). 8.3 Classical-quan tum corresp ondence for P VI Consider the P VI equation in the Calogero- lik e f orm (8.3) for a v aria ble u : ¨ u = 3 X k =0 ν k ℘ ′ ( u + ω k ) . (8.24) Recall that the v ariables u, t are connected with the original v ariables y , T in (8.1) b y the f orm ulas (8.1 6 ): y = ℘ ( u ) − e 1 e 2 − e 1 , T = e 3 − e 1 e 2 − e 1 (8.25) 35 and ν 0 = α = 2  ξ + 1 2  2 , ν 1 = − β = 2 ξ 2 0 , ν 2 = γ = 2 ξ 2 1 , ν 3 = 1 2 − δ = 2 ξ 2 2 . (8.26) This equation is equiv alen t to the zero curv ature condition for the matrices ˜ U ( x, t ), ˜ V ( x, t ) giv en by (8.15) with the elliptic pa r ametrization (8.1 6). The next step is a sp ecial dia gonal gauge tra nsformation { ˜ U , ˜ V } − → { U , V } of the form (2.9) that recasts the matrices in the form suc h that the condition b x = 2 B (2.4) is satisﬁed. As is sho wn in App endix C, the condition that the dep endence on x and u in t he gauge function ω factorizes is strong enough to ﬁx sim ultaneously t he elliptic substitution for the spectral parameter and the x -dep enden t par t of ω . The latter is found in the form ω 2 = ℘ ′ ( x ) ϑ 2 0 ( x ) 2( ℘ ( x ) − e 3 ) ρ 2 ( t ) , (8.27) where ρ ( t ) is some (y et unkno wn) function of t only (see (C14)). The function ρ is to b e determined at the very end from the condition tha t t he x -indep enden t part of the p oten tial in the non-stationary Schr¨ odinger equation b e equal to the classical Hamiltonian H VI ( ˙ u, u ). The detailed deriv atio n of ( 8 .27) is g iven in App endix C. Here w e can only sa y that if this express ion is known, then it is an easy exercise t o chec k that the x -deriv a tiv e of the r ig h t upp er elemen t b of the matrix U , b = ω 2 ˜ b = 2 K ( e 2 − e 1 ) ρ 2 ( t ) ϑ 2 0 ( x ) ℘ ( x ) − ℘ ( u ) ℘ ( x ) − e 3 , (8.28) app ears to b e equal t o 2 B = 2 ω 2 ˜ B . Therefore, in this gauge the non-stationary Schr¨ o- dinger equation of the f orm (2.19) do es hold. It remains to ﬁnd the p oten tial U ( x, t ) = 1 2 ( ad − bc − a x + 2 A ). T aking in to a ccoun t that a = ˜ a + ∂ x log ω , A = ˜ A + ∂ t log ω and ˜ a = ∂ X ∂ x a , ˜ A = ∂ T ∂ t A + ∂ X ∂ x ϑ ′ 0 ( x ) ϑ 0 ( x ) a , one can represen t it as a sum of three terms: U = U 1 + U 2 + U 3 , where U 1 = 1 2 ∂ X ∂ x ! 2 det U ( X ) = 1 2 ∂ X ∂ x ! 2 ( ad − b c ) U 2 = − 1 2   ∂ 2 X ∂ x 2 a + ∂ X ∂ x ! 2 a X   + ∂ T ∂ t A + ∂ X ∂ x ϑ ′ 0 ( x ) ϑ 0 ( x ) a − ∂ X ∂ x a ∂ x log ω U 3 = − 1 2 ( ∂ x log ω ) 2 − 1 2 ∂ 2 x log ω + ∂ t log ω . F or the purp ose of this pa per w e do not need the explicit form of the matrices U ( x, t ), V ( x, t ) in the elliptic parametrization (this will b e presen ted elsewhere). T ec hnically , it is con v enien t to mak e the calculations using the original v ar iables X , T where p ossible and pass to the elliptic parametrization a t the ve ry end. That is why w e hav e expressed the righ t hand sides in terms of the matrices U , V with rational dep endence on the sp ectral parameter X . 36 The calculation of the X -dep enden t part of U 1 is relat ively easy . The result is: U 1 = 2( e 2 − e 1 ) " − ξ 2 X − T ξ 2 0 X + ( T − 1) ξ 2 1 X − 1 − T ( T − 1) ξ 2 2 X − T # + U 1 , 0 = − 2 ξ 2 ( ℘ ( x ) − e 1 ) − 2 ξ 2 0 ( ℘ ( x + ω 1 ) − e 1 ) − 2 ξ 2 1 ( ℘ ( x + ω 2 ) − e 2 ) − 2 ξ 2 2 ( ℘ ( x + ω 3 ) − e 3 ) + U 1 , 0 . (8.29) The passage to the elliptic functions is done according to form ulas (8.20). Th e X - indep enden t part, U 1 , 0 , is U 1 , 0 = 2 ξ ( e 2 − e 1 ) h ( T + 1)( g 0 + ξ 0 ) + ( T − 1)( g 1 + ξ 1 ) − ( T − 1)( g 2 + ξ 2 ) i . (8.30) Using f o rm ulas from App endix A, we get: U 1 , 0 =  2 ξ 2 − 1 2  ( e 2 − e 1 ) y + 2( e 2 − e 1 ) T y − ( T − 1 ) 2 y 2 T 4 + ξ 2 0 ! + 2( e 2 − e 1 )( T − 1) y − 1 T 2 y 2 T 4 − ξ 2 1 ! + 2( e 2 − e 1 ) T ( T − 1) y − T − ( y T − 1) 2 4 + ξ 2 2 ! − ( e 2 − e 1 )( T − 1) 2 . (8.31) The T -deriv ative of y in the elliptic pa r ametrization reads y T = 1 2 T ( T − 1) ℘ ′ ( u ) ( e 2 − e 1 ) 2 ˙ u + ϑ ′ 0 ( u ) ϑ 0 ( u ) ! . (8.32) Plugging it to the right hand side of (8.31) and using form ulas (8.20) (no w with y , u instead of X, x ), w e obtain: U 1 , 0 =  2 ξ 2 − 1 2  ( ℘ ( u ) − e 1 ) + 2 ξ 2 0 ( ℘ ( u + ω 1 ) − e 1 ) + 2 ξ 2 1 ( ℘ ( u + ω 2 ) − e 2 ) +  2 ξ 2 2 − 1 2  ( ℘ ( u + ω 3 ) − e 3 ) − 1 2 ˙ u + ϑ ′ 0 ( u ) ϑ 0 ( u ) ! 2 + ℘ ′ ( u ) 2( e 2 − e 1 )( y − T ) ˙ u + ϑ ′ 0 ( u ) ϑ 0 ( u ) ! − e 3 − e 2 2 . (8.33) The un w an ted terms in the last line can b e transformed to logarithmic t -deriv ativ es using the f orm ulas ∂ t log ϑ 0 ( u ) = ˙ u ϑ ′ 0 ( u ) ϑ 0 ( u ) + 1 2 ϑ ′ 0 ( u ) ϑ 0 ( u ) ! 2 − 1 2 ℘ ( u + ω 3 ) − η (8.34) 1 2 ∂ t log( y − T ) = ℘ ′ ( u ) 2( e 2 − e 1 )( y − T ) ˙ u + ϑ ′ 0 ( u ) ϑ 0 ( u ) ! − ℘ ( u + ω 3 ) + e 3 (8.35) 37 (and thus they can b e eliminated by a proper c hoice of the function ρ ( t ), see b elo w). T aking this in to accoun t, w e obtain U 1 , 0 in the fo rm U 1 , 0 =  2 ξ 2 − 1 2  ( ℘ ( u ) − e 1 ) + 2 ξ 2 0 ( ℘ ( u + ω 1 ) − e 1 ) + 2 ξ 2 1 ( ℘ ( u + ω 2 ) − e 2 ) + 2 ξ 2 2 ( ℘ ( u + ω 3 ) − e 3 ) − ˙ u 2 2 + ∂ t log ( y − T ) 1 / 2 ϑ 0 ( u ) − e 3 + e 2 2 − η . (8.36) F or the calculation of U 2 w e prepare the formulas ∂ X ∂ x ∂ x log ω = 1 2 ∂ X ∂ x ∂ x log ℘ ′ ( x ) e 2 − e 1 e 2 − e 1 ℘ ( x ) − e 3 ! + ∂ X ∂ x ϑ ′ 0 ( x ) ϑ 0 ( x ) = 1 2 " ∂ 2 X ∂ x 2 − 1 X − T  ∂ X ∂ x  2 # + ∂ X ∂ x ϑ ′ 0 ( x ) ϑ 0 ( x ) ∂ 2 X ∂ x 2 a + 1 2  ∂ X ∂ x  2 a X = 2( e 2 − e 1 ) h 2 ξ X − ( T + 1)( g 0 + ξ 0 ) − T ( g 1 + ξ 1 ) − ( g 2 + ξ 2 ) i and 1 2  ∂ X ∂ x  2 a X − T + ∂ T ∂ t A = 2( e 2 − e 1 ) h ξ X − ( g 0 + ξ 0 ) + ( T − 1)( g 2 + ξ 2 ) i . The calculation giv es the follo wing simple result: U 2 = − 2 ξ ( ℘ ( x ) − e 3 ) . (8.37) A t last, let us ﬁnd U 3 . W e ha v e: ∂ t log ω = 1 2 ∂ t log ℘ ′ ( x ) ℘ ( x ) − e 3 ! + ∂ t log( ρϑ 0 ( x )) = 1 2 ∂ t log ∂ X ∂ x − 1 2 ∂ t log( X − T ) + ∂ t log( ρϑ 0 ( x )) = 1 2 ∂ X ∂ x ! − 1 ∂ 2 X ∂ x 2 + 1 / 2 X − T ∂ T ∂ t − ∂ X ∂ t ! + ∂ t log ϑ 0 ( x ) + ∂ t log ρ = 1 2 ∂ x log ℘ ′ ( x ) ℘ ( x ) − e 3 ! ∂ x log ϑ 0 ( x ) + 1 2 ( ∂ x log ϑ 0 ( x )) 2 + ∂ t log ρ − e 3 − 2 η , where η = − 1 6 ϑ ′′′ 1 (0) ϑ 1 (0) . When pa ss ing to the la st line w e ha v e used the heat equation (8.21) and the relation ∂ 2 x log ϑ 0 ( x ) = − ℘ ( x + ω 3 ) − 2 η . Com bining the diﬀeren t contributions to U 3 and passing to t he elliptic parametrization, we ﬁnd: U 3 = − 1 8  3 ℘ ( x ) − ℘ ( x + ω 1 ) − ℘ ( x + ω 2 ) − ℘ ( x + ω 3 )  − e 3 2 − η + ∂ t log ρ . (8.38) 38 Using the form ulas giv en ab ov e and equation (A13), w e obtain the p oten tial in the form U ( x, t ) = −  2  ξ + 1 2  2 − 1 8  ℘ ( x ) −  2 ξ 2 0 − 1 8  ℘ ( x + ω 1 ) −  2 ξ 2 1 − 1 8  ℘ ( x + ω 2 ) −  2 ξ 2 2 − 1 8  ℘ ( x + ω 3 ) − ˜ H , (8.39) where ˜ H = ˙ u 2 2 − 3 X k =0 ν k ℘ ( u + ω k ) + 1 2 ∂ t log   ϑ 2 0 ( u )( ϑ ′ 1 (0)) 8 3 ( y − T ) K ( T ) ϑ 6 0 (0) ρ 2 ( t )   with the same ν k as in (8.26) . Using the iden tities from App endix B one can expres s y − T in terms of the theta-functions: 1 y − T = e 2 − e 1 ℘ ( u ) − e 3 = − π 2 ϑ 6 0 (0) ϑ 2 1 ( u ) ( ϑ ′ 1 (0)) 2 ϑ 2 0 ( u ) . Therefore, c ho osing ρ ( t ) = ( ϑ ′ 1 (0)) 1 3 ϑ 1 ( u ) q K ( T ) , w e see that ˜ H = H ( α,β ,γ ,δ ) VI ( ˙ u, u ) with the same parameters as in (8 .1 3). With this c hoice of ρ , the gauge function ω (8.27 ) acquires the form ω 2 = ( ϑ ′ 1 (0)) 5 3 ϑ 0 (0) ϑ 2 (0) ϑ 3 (0) ϑ 2 ( x ) ϑ 3 ( x ) ϑ 0 ( x ) ϑ 2 1 ( u ) ϑ 1 ( x ) K ( T ) . (8.40) Finally , we conclude that the classical-quan tum corresp ondence do es w ork f o r the P VI equation. The non-stationary Sc hr¨ odinger equation is ∂ t ψ = h H ( α − 1 8 ,β + 1 8 ,γ − 1 8 ,δ + 1 8 ) VI ( ∂ x , x ) − H ( α,β ,γ ,δ ) VI ( ˙ u, u ) i ψ , (8.41) where H ( α − 1 8 ,β + 1 8 ,γ − 1 8 ,δ + 1 8 ) VI ( ∂ x , x ) = 1 2 ∂ 2 x − 3 X k =0  ν k − 1 8  ℘ ( x + ω k ) and the pa rameters ν k are connected with α , β , γ , δ as in (8.26). The function Ψ( x, t ) = e R t H ( α, β , γ , δ ) VI ( ˙ u,u ) dt ′ ψ ( x, t ) (8.42) th us ob eys the non-statio na ry Sc hr¨ odinger equation ∂ t Ψ = H ( α − 1 8 , β + 1 8 , γ − 1 8 , δ + 1 8 ) VI ( ∂ x , x )Ψ =  1 2 ∂ 2 x + V ( α − 1 8 , β + 1 8 , γ − 1 8 , δ + 1 8 ) VI ( x, t )  Ψ . (8.43) Note that in the quantum part all the parameters undergo shifts b y ± 1 8 . In terms of the parameters ν k (see (8.3), (8.2 6)) the shifts a r e ν k → ν k − 1 8 , k = 0 , . . . , 3. In particular, if all ν k are equal to each other, ν k = ν , then w e obtain the non-stationary Lam ´ e equation ∂ t Ψ =  1 2 ∂ 2 x − 4 ˜ ν ℘  2 x    1 , 2 π it  Ψ , ˜ ν = ν − 1 8 . (8.44) 39 (the iden tity 3 P k =0 ℘ ( x + ω k ) = 4 ℘ (2 x ) has b een used). W e remark that the non- stationary Lam ´ e equation in connection with the P VI equation (and with the 8-vertex mo del) w as discusse d in [42]. Recen tly , the non- stationary Lam´ e equation has app eared [43, 44] in the context of the A GT conjecture. T o summarize, similar to t he other cases, w e hav e tw o equiv alen t represen tat io ns of the P VI equation. One is a classic al mot io n in the time-dep enden t p erio dic p oten tial with Hamiltonian (8.4). The co ordinate o f the particle as a function of time ob eys t he P VI equation. Another represen tation is a quan tum mec hanical particle in the time- dep enden t p oten tial of the same form with mo diﬁed co eﬃcien ts describ ed b y a non- stationary Sc hr¨ odinger equation. The latter sim ultaneously serv es as the linear problem for t ime ev olution asso ciated with the Painle v ´ e equation. 9 Conclud ing remarks W e hav e show n that for each P a inlev ´ e equation written in the “Calogero fo rm” ¨ u = − ∂ u V ( u, t ) with a time-dep enden t p oten tial V ( x, t ), the asso ciated linear problems can b e represen ted a s             1 2 ∂ 2 x − 1 2 ( ∂ x log b ( x, t )) ∂ x + ˜ W ( x, t )  Ψ = E Ψ ∂ t Ψ =  1 2 ∂ 2 x + ˜ V ( x, t )  Ψ , (9.1) The second equation is the non-stationary Schr¨ odinger e quation in imagin ary time wi th the p otential ˜ V ( x, t ) that has the same form as the classic al p otential f o r the Painlev´ e e quation (with p ossibly mo diﬁed parameters). The p otential in the ﬁrst equation is ˜ W ( x, t ) = ˜ V ( x, t ) − ∂ t b ( x, t ) 2 b ( x, t ) + ∂ 2 x b ( x, t ) 4 b ( x, t ) and the eigen v alue E is the v alue of the classical Hamiltonian H ( ˙ u, u ) f o r the P ainlev ´ e equation in the Calogero fo rm (with the o pp osite sign): E = − H ( ˙ u, u ) = − ˙ u 2 2 − V ( u, t ) . These equations has b een derived fr o m the 2 × 2 matrix linear pro blems (1.3) with the matrices U ( x, t ), V ( x, t ) of the sp ecial form, with the function b ( x , t ) b eing the right upp er en try of the matrix U ( x, t ). The second equation o f the sy stem (9.1) describ es isomono dromic defor mations of the ﬁrst one a nd their compatibilit y implies the P ainlev ´ e equation (in the Calogero f orm) fo r the function u = u ( t ) deﬁned implicitly as zero of the f unction b ( x, t ): b ( u ( t ) , t ) = 0. In short, the conclus ion is that linearization o f the P ainlev ´ e equation, i.e., passing to the linear problem, is equiv alen t to its quan tization. The imaginary time suggests an in terpretation in terms of the F okk er-Planc k equation for a sto c hastic pro cess. 40 Here a remark is in order. On the one hand, the P ainlev ´ e eq uation is obtained as a compatibilit y condition for the pair of equations (9.1 ). How ev er, on the other hand, the second equation alone is already enough to encode the full information ab out the P ainlev ´ e equation. Indeed, it describ es a quan tum mec hanical pa r ticle on the line in the time-dep enden t p oten tial corresp onding to the Painlev ´ e equation. Therefore, the P ainlev ´ e equation itself should emerge in the classical limit. In the pap ers [45, 46] the K nizhnik -Za molo dchik ov system of equations w as treated as a natural quantization of isomono dromic deformations. It w ould b e v ery in teresting to understand o ur results in these t erms. A t last, we w ould lik e to p oin t out that another sort of classical-quan t um corresp on- dence for P ainlev ´ e equations w as established in the w ork [47]. Namely , it w as sho wn that each equation from the Painlev ´ e list could b e rega rded as a “classical analog” of a linear ordinary diﬀeren tial equation of the Heun class in the sense that the second-order diﬀeren tial op erator ˆ L ( ∂ x , x ) in v olv ed in the la tter, after a prop erly tak en classical limit, coincides with the p olynomial classical Hamiltonian f or the Painlev ´ e equation. (In o ther w ords, the Euler-Lagrange equation corresponding to the sym b ol L ( p, q ) of the linear diﬀeren tial o perato r ˆ L is j ust the P ainlev ´ e equation.) Similarly to our approach, in this classical/quan tum mechanic al interpretation, the time v ariable T has the meaning of the deformation parameter. How ev er, the imp ortan t diﬀerence is that [47] deals with sta- tionary Schr¨ odinger-lik e equation with co eﬃcien ts dep ending on T . It seems t o us that the construction elabo r ated in the presen t pap er a nd a similar construction suggeste d previously in [30] are more appropriate b ecause the Painlev ´ e equations a re essen tially non-autonomous systems and it is really natural to asso ciate no n -stationary Sch r¨ odinger equations with them. Ac kno wledgmen ts The authors a re grateful to I.Kric hev er, S.Oblezin and V.P ob erezhniy for discus sions. They also thank V.P ob erezhniy and B.Suleimanov f o r bringing the pap ers [47] and [30, 31, 32] to t heir atten t io n. The work of b oth authors w as partially supp orted b y Russian F ederal Nuclear Energy Agency under con tract H.4e.45.90.1 1.1059. The work of A.Zabro din was supp orted in part b y RFBR grant 1 1-02-01220 , by j o in t RFBR grants 09-01- 9 3106-CNRS, 10 -02-92109- JSPS and by the F ederal Agency for Science and Inno- v ations of Russian F ederation under con tract 14.74 0.11.0081. The w ork of A.Zotov w as supp orted in part b y gran ts R FBR-09-02- 0 0393, RF BR -09-01-924 37-KEa, RF BR-09-01- 93106-CNRS, Russian Preside nt fund MK- 1 646.2011.1 and b y the F ederal Agency for Science and Innov ations of R uss ian F ederation under con tract 14.7 40.11.0347. App en d ix A In this app endix we presen t some details of t he deriv at ion of the P VI equation from t he zero curv ature condition f or matrices (8.7), (8.8 ) . W e use the notatio n in tro duced in the main text. 41 First o f all, let us write down t he diﬀeren tial equations for the functions g i , u i that follo w from the zero curv atur e condition. The full system of equations reads T ∂ T ( u 0 g 0 ) = 2 u 0 g 0 ( g 0 + g 2 + ξ 0 + ξ 2 ) + 2 u 1 g 1 ( g 0 + ξ 0 ) ( T − 1) T ∂ T ( u 1 g 1 ) = 2 u 1 g 1 ( g 1 + g 2 + ξ 1 + ξ 2 ) + 2 u 0 g 0 ( g 1 + ξ 1 ) T ∂ T g 0 = u 0 u 2 g 0 ( g 2 + 2 ξ 2 ) − u 2 u 0 g 2 ( g 0 + 2 ξ 0 ) ( T − 1 ) ∂ T g 1 = u 1 u 2 g 1 ( g 2 + 2 ξ 2 ) − u 2 u 1 g 2 ( g 1 + 2 ξ 1 ) T ∂ T  g 0 + 2 ξ 0 ) u 0  = 2 u 0 ( g 0 + 2 ξ 0 )( g 2 + ξ 2 ) − 2 u 2 ( g 2 + 2 ξ 2 )( g 0 + ξ 0 ) ( T − 1 ) ∂ T  g 1 + 2 ξ 1 ) u 1  = 2 u 1 ( g 1 + 2 ξ 1 )( g 2 + ξ 2 ) − 2 u 2 ( g 2 + 2 ξ 2 )( g 1 + ξ 1 ) (A1) along with the integrated relations (8.9). How ev er, a direct deriv ation of t he P VI equation from this system is not the easiest w a y . Belo w w e giv e a short-cut whic h closely follows the deriv ation outlined in [1 8]. Along with the function y deﬁned b y (8.10) let us also introduce the function z = a ( y ) = g 0 + ξ 0 y + g 1 + ξ 1 y − 1 + g 2 + ξ 2 y − T . (A2) Then, from the fact that the to tal T - deriv a tiv e of b ( y ) is zero, w e write, using the zero curv ature equations in the f orm (2.3): 0 = d b ( y ) /dT = b X ( y ) y T + b T ( y ) = b T ( y ) y T + B X ( y ) − 2 z B ( y ) = 0, where y T ≡ dy /dT . Expres sing b X ( y ), etc in terms of the functions K and y (see (8.11)), we obtain: y T = y ( y − 1)( y − T ) T ( T − 1)  2 z + 1 y − T  . (A3) Com bining the integrals of motion (8 .9 ) with the deﬁnition of z , and using f orm ulas (8.12), w e can write t he system of equations                        g 0 + g 1 + g 2 = ξ 3 g 0 + ξ 0 y + g 1 + ξ 1 y − 1 + g 2 + ξ 2 y − T = z T g 0 ( g 0 + 2 ξ 0 ) y − ( T − 1) g 1 ( g 1 + 2 ξ 1 ) y − 1 + T ( T − 1) g 2 ( g 2 + 2 ξ 2 ) y − T = 0 (A4) for t he three functions g i whic h can b e solve d a s g 0 = − y 2 ξ T h y ( y − 1)( y − T ) ˜ z 2 − 2  ξ 3 ( y − 1)( y − T ) − ξ 1 ( y − T ) − ξ 2 T ( y − 1)  ˜ z + ξ 3  ξ 3 ( y − 1) − (2 ξ 2 + ξ 3 ) T − 2 ξ 1 i (A5) 42 g 1 = y − 1 2 ξ ( T − 1) h y ( y − 1)( y − T ) ˜ z 2 − 2  ξ 3 y ( y − T ) + ξ 0 ( y − T ) − ξ 2 ( T − 1) y  ˜ z + ξ 3  ξ 3 ( y − 1) − (2 ξ 2 + ξ 3 )( T − 1) + 2 ξ 0 + ξ 3 i (A6) g 2 = − y − T 2 ξ T ( T − 1) h y ( y − 1)( y − T ) ˜ z 2 − 2  ξ 3 y ( y − 1) + ξ 0 T ( y − 1) + ξ 1 ( T − 1) y  ˜ z + ξ 3  ξ 3 ( y − 1) + (2 ξ 0 + ξ 3 ) T + 2 ξ 1 ( T − 1 ) i , (A7) where ˜ z = z − ξ 0 y − ξ 1 y − 1 − ξ 2 y − T = g 0 y + g 1 y − 1 + g 2 y − T (A8) and ξ = ξ 0 + ξ 1 + ξ 2 + ξ 3 . In order to ﬁnd a more explicit represen tation, we notice that the functions g 0 /y , g 1 / ( y − 1) a nd g 2 / ( y − T ) are rational functions of the v ariable y with ﬁrst order p oles at 0, 1 , T a nd ∞ . Calculating the residues, one can write them in t he explicit form: 2 ξ g 0 y = − ( ξ + 1 2 ) 2 T y − G 0 T − 1 y " ( T − 1 ) 2 4 y 2 T − ξ ( T − 1) y T + ξ 0 (2 ξ − ξ 0 ) # + T − 1 T ( y − 1) " T 2 4 y 2 T − ξ 2 1 # − T − 1 y − T  1 4 ( y T − 1) 2 − ξ 2 2  (A9) 2 ξ g 1 y − 1 = ( ξ + 1 2 ) 2 T − 1 y + G 1 T − 1 + T ( T − 1) y " ( T − 1) 2 4 y 2 T − ξ 2 0 # + 1 y − 1 " − T 2 4 y 2 T − ξ T y T + ξ 1 ( ξ 1 − 2 ξ ) # + T y − T  1 4 ( y T − 1) 2 − ξ 2 2  (A10) 2 ξ g 2 y − T = − ( ξ + 1 2 ) 2 T ( T − 1) y − G 2 T ( T − 1) − 1 ( T − 1) y " ( T − 1 ) 2 4 y 2 T − ξ 2 0 # + 1 T ( y − 1) " T 2 4 y 2 T − ξ 2 1 # − 1 y − T  1 4 ( y T − 1) 2 − ξ ( y T − 1) + ξ 2 (2 ξ − ξ 2 )  (A11) and G 0 = T − 1 4 − ξ ( ξ T + ξ + 1) G 1 = T − 1 4 − ξ 2 ( T − 1 ) G 2 = T − 1 4 + ξ ( ξ + 1)( T − 1) . (A12) 43 The next step is to express the T - deriv ativ e of the function z in terms of the functions g i and y . F or that purp ose, we write z T = a X ( y ) y T + a T ( y ) and use the zero curv ature equation a T ( X ) − A X ( X ) + b ( X ) C ( X ) − c ( X ) B ( X ) = 0 to obtain z T = a X ( y ) y T + A X ( y ) + c ( y ) B ( y ) = − g 0 + ξ 0 y 2 + g 1 + ξ 1 ( y − 1) 2 + g 2 + ξ 2 ( y − T ) 2 ! y T + g 2 + ξ 2 ) ( y − T ) 2 + 1 T ( T − 1) T g 0 ( g 0 + 2 ξ 0 ) y 2 − ( T − 1) g 1 ( g 1 + 2 ξ 1 ) ( y − 1) 2 + T ( T − 1) g 2 ( g 2 + 2 ξ 2 ) ( y − T ) 2 ! . It remains to express z T in terms of y , y T , y T T with the help of (A3) and to plug the explicit form of the functions g i giv en b y equations (A9)–(A11). After a long calculation, one obtains the P VI equation (8.1) with the parameters (8.1 3). In establishing the classical-quan tum correspondence w e also need the T -deriv ative of t he function K ( T ). A straightforw ard calculation, whic h uses formulas (8.12) and the ﬁrst tw o equations o f the system (A1), yields: ∂ T log K = − (2 ξ + 1) y − T T ( T − 1) . (A13) App en d ix B Theta-functions, W ei erstrass ℘ -fun c tion and other useful func- tions Theta-functions. The Jacobi’s theta-functions ϑ a ( z ) = ϑ a ( z | τ ), a = 0 , 1 , 2 , 3, are deﬁned b y t he formulas ϑ 1 ( z ) = − X k ∈ Z exp  π iτ ( k + 1 2 ) 2 + 2 π i ( z + 1 2 )( k + 1 2 )  , ϑ 2 ( z ) = X k ∈ Z exp  π iτ ( k + 1 2 ) 2 + 2 π iz ( k + 1 2 )  , ϑ 3 ( z ) = X k ∈ Z exp  π iτ k 2 + 2 π iz k  , ϑ 0 ( z ) = X k ∈ Z exp  π iτ k 2 + 2 π i ( z + 1 2 ) k  , (B1) where τ is a complex par a meter (the mo dular parameter) suc h that Im τ > 0. The func- tion ϑ 1 ( z ) is o dd, t he other three functions are eve n. The inﬁnite pr o duct represen tation for t he ϑ 1 ( z ) reads: ϑ 1 ( z ) = i exp  iπ τ 4 − iπ z  ∞ Y k =1  1 − e 2 π ikτ  1 − e 2 π i (( k − 1) τ + z )  1 − e 2 π i ( kτ − z )  . (B2) 44 In order to unify some formulas g iv en b elo w, it is con v enien t to understand the index a mo dulo 4 , i.e., to identify ϑ a ( z ) ≡ ϑ a +4 ( z ). Set ω 0 = 0 , ω 1 = 1 2 , ω 2 = 1 + τ 2 , ω 3 = τ 2 , then the f unction ϑ a ( z ) has simple zeros at the p oin ts of the lattice ω a − 1 + Z + Z τ (here ω a ≡ ω a +4 ). The theta-functions hav e the fo llo wing quasi-p erio dic prop erties under shifts b y 1 and τ : ϑ a ( z + 1) = e π i (1+ ∂ τ ω a − 1 ) ϑ a ( z ) ϑ a ( z + τ ) = e π i ( a + ∂ τ ω a − 1 ) e − π iτ − 2 π iz ϑ a ( z ) . (B3) Shifts by the half- perio ds relate t he diﬀeren t theta-functions to each ot her: ϑ 1 ( z + ω 1 ) = ϑ 2 ( z ) , ϑ 3 ( z + ω 1 ) = ϑ 0 ( z ) , (B4) ϑ 1 ( z + ω 2 ) = e − πiτ 4 − π iz ϑ 3 ( z ) , ϑ 2 ( z + ω 2 ) = − ie − πiτ 4 − π iz ϑ 0 ( z ) (B5) ϑ 1 ( z + ω 3 ) = ie − πiτ 4 − π iz ϑ 0 ( z ) , ϑ 2 ( z + ω 3 ) = e − πiτ 4 − π iz ϑ 3 ( z ) . (B6) W eierstrass ℘ -function. The W eierstrass ℘ -function can b e deﬁned by the formula ℘ ( z ) = − ∂ 2 z log ϑ 1 ( z ) − 2 η , (B7) where η = − 1 6 ϑ ′′′ 1 (0) ϑ ′ 1 (0) = − 2 π i 3 ∂ τ log θ ′ 1 (0 | τ ) . (B8) The function ℘ ( z ) is double-p erio dic with p erio ds 2 ω 1 = 1, 2 ω 3 = τ , ℘ ( z + M + N τ ) = ℘ ( z ), M , N ∈ Z , and ha s second order p oles at the origin (and at all the p oin ts M + N τ with integer M , N ). The deriv ativ e of the ℘ - function is g iv en by ℘ ′ ( z ) = − 2 ( ϑ ′ 1 (0)) 3 ϑ 2 (0) ϑ 3 (0) ϑ 0 (0) ϑ 2 ( z ) ϑ 3 ( z ) ϑ 0 ( z ) ϑ 3 1 ( z ) . (B9) The v alues of the ℘ -function at the half-p erio ds, ω k , e 1 = ℘ ( ω 1 ) , e 1 = ℘ ( ω 2 ) , e 3 = ℘ ( ω 3 ) (B10) pla y a sp ecial role. The sum of the num b ers e k is zero: e 1 + e 2 + e 3 = 0. The diﬀerences e j − e k can b e represen ted in terms of the v alues of the theta-functions a t z = 0 (t heta- constan ts) in t w o diﬀeren t wa ys: e 1 − e 2 = π 2 ϑ 4 0 (0) = 4 π i ∂ τ log ϑ 3 (0) ϑ 2 (0) e 1 − e 3 = π 2 ϑ 4 3 (0) = 4 π i ∂ τ log ϑ 0 (0) ϑ 2 (0) e 2 − e 3 = π 2 ϑ 4 2 (0) = 4 π i ∂ τ log ϑ 0 (0) ϑ 3 (0) . (B11) 45 The second represen ta t io n is a consequence o f the heat equation (B30) (see b elo w). Its another consequence is a represen tation o f the e k ’s themselv es as logarit hmic τ -deriv ativ es of the theta-constants: e k = 4 π i ∂ τ  1 3 log ϑ ′ 1 (0) − log ϑ k +1 (0)  . (B12) Using the ﬁrst equalities in ( B1 1) and the heat equation, the τ -deriv ativ es of the diﬀer- ences e j − e k can b e expressed through the e k ’s a nd η as follows: π i ∂ τ log( e j − e k ) = − e l − 2 η . (B13) Here { j k l } stands fo r an y cyclic p erm utat io n of { 123 } . Subtracting t w o suc h equations, w e also g et π i ∂ τ log e j − e k e l − e k = e j − e l . (B14) The ℘ -function ob eys the diﬀeren tial equation ( ℘ ′ ( z )) 2 = 4( ℘ ( z ) − e 1 )( ℘ ( z ) − e 2 )( ℘ ( z ) − e 3 ) . (B15) W e also men tion the f orm ulae ℘ ( z ) − e k = ( ϑ ′ 1 (0)) 2 ϑ 2 k +1 (0) ϑ 2 k +1 ( z ) ϑ 2 1 ( z ) . (B16) Eisenstein functions and Φ -function. Sometimes it is con v enien t to use the Eisen- stein functions E 1 ( z ) = ∂ z log ϑ 1 ( z ) , E 2 ( z ) = − ∂ z E 1 ( z ) = − ∂ 2 z log ϑ 1 ( z ) = ℘ ( z ) + 2 η . (B17) The function E 1 is quasi-p erio dic, E 1 ( z + 1) = E 1 ( z ), E 1 ( z + τ ) = E 1 ( z ) − 2 π i , while E 2 is double-p erio dic: E 2 ( z + 1) = E 2 ( z ), E 2 ( z + τ ) = E 2 ( z ). Near z = 0 they hav e the expansions E 1 ( z ) = 1 z − 2 η z + . . . , E 2 ( z ) = 1 z 2 + 2 η + . . . It is not diﬃcult to see that the function E 1 ( z ) has the following v alues at the half-p erio ds: E 1 ( ω j ) = − 2 π i∂ τ ω j (B18) and, therefore, the identit y E 1 ( ω j ) + E 1 ( ω k ) = E 1 ( ω j + ω k ) (B19) holds t r ue for any diﬀeren t j, k = 1 , 2 , 3. The follo wing function app ears to b e useful in the calculations: Φ( u, z ) = ϑ 1 ( u + z ) ϑ ′ 1 (0) ϑ 1 ( u ) ϑ 1 ( z ) . (B20) It o b eys the ob vious prop erties Φ( u, z ) = Φ( z , u ), Φ( − u, − z ) = − Φ( u, z ) as w ell as less ob vious ones: Φ( u, z )Φ( − u, z ) = ℘ ( z ) − ℘ ( u ) (B21) 46 Φ( u, z )Φ( w , z ) = Φ( u + w , z )( E 1 ( z ) + E 1 ( u ) + E 1 ( w ) − E 1 ( z + u + w )) . (B22) Here E 1 ( z ) is the ﬁrst Eisenstein function. The expansion of the function Φ( u, z ) near z = 0 is Φ( u, z ) = 1 z + E 1 ( u ) + z 2 ( E 2 1 ( u ) − ℘ ( u )) + O ( z 2 ) . (B23) The quasi-p erio dicit y pro perties of the function Φ are: Φ( u, z + 1) = Φ( u, z ) , Φ( u, z + τ ) = e − 2 π iu Φ( u, z ) . (B24) The z -deriv ative of the function Φ is equal to ∂ z Φ( u, z ) = Φ( u, z )( E 1 ( u + z ) − E 1 ( z )) . (B25) Finally , let us in tro duce the functions ϕ j ( z ) = e 2 π iz∂ τ ω j Φ( z , ω j ) , j = 1 , 2 , 3 . (B26) Setting u in (B21) and (B22) to b e equal to the half-p erio ds, we ha v e: ϕ 2 j ( z ) = ℘ ( z ) − e j , ϕ 2 j ( z ) − ϕ 2 k ( z ) = e k − e j (B27) ϕ j ( z ) ϕ k ( z ) = ϕ l ( z )( E 1 ( z ) + E 1 ( ω l ) − E 1 ( z + ω l )) . (B28) In a similar w ay , fr o m (B25) and (B18) it fo llo ws that ∂ z ϕ j ( z ) = ϕ j ( z ) h E 1 ( z + ω j ) − E 1 ( ω j ) − E 1 ( z ) i = − ϕ k ( z ) ϕ l ( z ) , (B29) where j, k , l is an y cyclic p ermu tation o f 1 , 2 , 3. Heat equation and related form u lae As it can b e easily seen from the deﬁnition (B1), a ll the theta- functions satisfy the “heat equation” 4 π i∂ τ ϑ a ( z | τ ) = ∂ 2 z ϑ a ( z | τ ) (B30) or, in terms of t he v ariable t = τ 2 π i used in the main text, 2 ∂ t ϑ a ( z ) = ∂ 2 z ϑ a ( z ). One can also in tro duce the “heat co eﬃcien t” κ = 1 2 π i and rewrite the heat equation in the form ∂ τ ϑ a ( z | τ ) = κ 2 ∂ 2 z ϑ a ( z | τ ) . All formulas for deriv ative s of elliptic functions with resp ect to the mo dular parameter ar e based on the heat equation. The τ -deriv atives of the functions Φ, E 1 and E 2 are giv en b y the following prop osition. Prop osition 1 The i d entities ∂ τ Φ( z , u ) = κ∂ z ∂ u Φ( z , u ) , (B31) ∂ τ E 1 ( z ) = κ 2 ∂ z ( E 2 1 ( z ) − ℘ ( z )) , (B32) ∂ τ E 2 ( z ) = κE 1 ( z ) E ′ 2 ( z ) − κE 2 2 ( z ) + κ 2 ℘ ′′ ( z ) , (B33) with the “h e at c o eﬃcient” κ = 1 2 π i , h old true 3 . 3 (B31) was obta ined in [25],[35]. 47 Pr o of : First w e prov e (B31). It follow s from (B30) that 4 π i ∂ τ ϑ 1 ( z ) ϑ 1 ( z ) = ϑ ′′ 1 ( z ) ϑ 1 ( z ) = ∂ z ϑ ′ 1 ( z ) ϑ 1 ( z ) ! + ϑ ′ 1 ( z ) ϑ 1 ( z ) ! 2 = − E 2 ( z ) + E 2 1 ( z ) . (B34) Therefore, ∂ τ Φ( z , u ) = κ 2  − 6 η − E 2 ( z + u ) + E 2 1 ( z + u ) + E 2 ( z ) − E 2 1 ( z ) + E 2 ( u ) − E 2 1 ( u )  , (B35) where the constant η is g iv en b y (B8). On the other hand, ∂ z ∂ u Φ( z , u ) = ∂ z h Φ( z , u )( E 1 ( z + u ) − E 1 ( u )) i = Φ( z , u ) × h ( E 1 ( z + u ) − E 1 ( u ))( E 1 ( z + u ) − E 1 ( z )) − E 2 ( z + u ) i . (B36) The rest o f the pro of is a direct use of the identit y ( E 1 ( z + u ) − E 1 ( u ) − E 1 ( z )) 2 = ℘ ( z ) + ℘ ( u ) + ℘ ( z + u ) . (B37) Equation (B32) easily follow s from (B31) and the lo cal expansion (B23) a round u = 0. Equation (B33) is just a deriv ative of (B32). Next let us prov e (8 .22) 4 . Prop osition 2 Set X ( z ) = ℘ ( z ) − e 1 e 2 − e 1 , then ∂ τ X = κ ∂ z X ∂ z log θ 0 ( z ) . (B38) Pr o of : The τ -deriv ative of X ( z ) = ℘ ( z ) − e 1 e 2 − e 1 ( B27 ) = ϕ 2 1 ( z ) e 2 − e 1 is: ∂ τ X = 2 ϕ 1 ( z ) ∂ τ ϕ 1 ( z )( e 2 − e 1 ) − ∂ τ ( e 2 − e 1 ) ϕ 2 1 ( z ) ( e 2 − e 1 ) 2 . Using t he deﬁnition of ϕ 1 ( z ) a nd the “heat equation” (B31) for the Φ-function, we write ∂ τ ϕ 1 ( z ) ( B25 ) = κ∂ z h ϕ 1 ( z )( E 1 ( z + ω 1 ) − E 1 ( ω 1 )) i = κ∂ z h ϕ 1 ( z ) E 1 ( z + ω 1 ) i = κ∂ z ϕ 1 ( z ) E 1 ( z + ω 1 ) − κϕ 1 ( z ) E 2 ( z + ω 1 ) . (B39) Substituting this and ∂ τ ( e 2 − e 1 ) ( B13 ) = − 2 κ ( e 2 − e 1 ) E 2 ( e 3 ) into (B39), w e hav e: ∂ τ X = 2 κ e 2 − e 1  ϕ 1 ( z ) ∂ z ϕ 1 ( z ) E 1 ( z + ω 1 ) − ϕ 2 1 ( z ) E 2 ( z + ω 1 ) + E 2 ( ω 3 ) ϕ 2 1 ( z )  . (B4 0) 4 This formula was prov ed by K .T ak asaki in [2 9] by comparis o n of a nalytic prop erties of the b oth sides. Here we give another pro of by a direct computation. 48 Since ∂ z X = 2 ϕ 1 ( z ) ∂ z ϕ 1 ( z ) e 2 − e 1 , w e can rewrite the latter equation as ∂ τ X = κ∂ z X E 1 ( z + ω 1 ) + 2 κϕ 2 1 ( z ) e 2 − e 1 ( − E 2 ( z + ω 1 ) + E 2 ( ω 3 )) whic h can b e furt her simpliﬁed with the help of the identit y E 2 ( z + ω 1 ) = E 2 ( ω 1 ) + ( e 2 − e 1 )( e 3 − e 1 ) ϕ 2 1 ( z ) . Dividing b oth sides b y ∂ z X , w e get ∂ τ X ∂ z X = κE 1 ( z + ω 1 ) + 2 κ ( e 3 − e 1 ) X − 1 ∂ z X . (B41) The last term can b e tra nsformed using the identities e 3 − e 1 ( B27 ) = ϕ 2 1 ( z ) − ϕ 2 3 ( z ), ∂ z X ( B29 ) = − 2 ϕ 1 ( z ) ϕ 2 ( z ) ϕ 3 ( z ) e 2 − e 1 and X − 1 ( B27 ) = ϕ 2 2 ( z ) e 2 − e 1 : ∂ τ X = κ∂ z X E 1 ( z + ω 1 ) + ϕ 2 ( z ) ϕ 3 ( z ) ϕ 1 ( z ) − ϕ 1 ( z ) ϕ 2 ( z ) ϕ 3 ( z ) ! . (B42) Finally , the desired form ula (B38) is obtained from this using (B2 8): ∂ τ X = κ∂ z X ( E 1 ( z + ω 3 ) − E 1 ( ω 3 )) = κ∂ z X ∂ z log θ 0 ( z ) . (B43) App en d ix C Gauge transformation of the linear p roblems for P VI In the para metrization (8.10), (8.12), the upp er right en tries of t he matrices U ( X, T ), V ( X , T ) forming the mo diﬁed Jim b o-Miw a U – V pair for the P VI equation are U 12 = b = K ( X − y ) X ( X − 1 )( X − T ) , V 12 = B = K ( y − T ) T ( T − 1)( X − T ) . P assing to a parametrization X = X ( x, t ), T = T ( t ) according to the rule (2.27) and p erforming a diagonal g a uge transformation of the form (2.9) we get the follow ing ex- pressions for the upp er right entries of the matrices U ( x, t ), V ( x, t ): b = U 12 = b X x ω 2 = K ( X − y ) X ( X − 1)( X − T ) X x ω 2 , (C1) B = V 12 = ( T t B + X t b ) ω 2 = K ( y − T ) T ( T − 1)( X − T ) T τ ω 2 + K ( X − y ) X ( X − 1)( X − T ) X t ω 2 . (C 2) 49 The x -deriv ativ e of b is b x = ( X − y ) ω 2 X ( X − 1)( X − T ) f + X 2 x X − y ! , (C3) where the not ation f = X xx + X x ∂ x log( ω 2 ) − X 2 x  1 X + 1 X − 1 + 1 X − T  is intro duced for brevit y . F urther, let us imp ose condition of the form (2.4): b x = k B , (C4) with some constan t k (not y et ﬁxed). Substituting (C2) and (C3) in to (C4), we obtain an equalit y of tw o linear functions of y pr ovid e d ∂ x log ω do es no t dep end on y . (The la tter assumption is necessary to achie v e separation of the v ar ia bles x , u in the non-stationa r y Sc hr¨ odinger equation.) Assuming this, we equate the co eﬃcien ts in front of y and the y -indep enden t terms in the b oth sides and get the system of t w o equations              f = k X t − X ( X − 1 ) T ( T − 1) T t ! X f + ( X x ) 2 = k X  X t − X − 1 T − 1 T t  . (C5) from whic h the functions X ( x, t ) and ∂ x log ω can b e determined. Excluding f , w e arrive at the diﬀeren tial equation for X : X 2 x = k T t T ( T − 1) X ( X − 1)( X − T ) . (C6) W e kno w that T t is g iven by (8.23): T t = 2( e 2 − e 1 ) T ( T − 1) . Therefore, X 2 x = 2 k ( e 2 − e 1 ) X ( X − 1)( X − T ) . (C7) This relation prompts the elliptic parametrization (8 .16) and ﬁxes the v alue of k : k = 2 . (C8) Note that in some sense this is “the same” co eﬃcien t 2 that enters the heat equation for theta-functions in the t -v a riable t = κτ : 2 ∂ t ϑ a ( x ) = ∂ 2 x ϑ a ( x ). In the same sense the non-stationary Sch r¨ odinger equation for the ψ - function is a “dressed” ve rsion of the heat equation. No w w e ar e ready to ﬁx the x -dep enden t part of the function ω 2 . F rom the ﬁrst equation of the system (C5) w e ﬁnd: ∂ x log ω 2 = − X xx X x + X x  1 X + 1 X − 1 + 1 X − T  − k X ( X − 1 ) T ( T − 1) T t X x + k X t X x . (C9) It is easy to sho w that X xx X x = X x 2  1 X + 1 X − 1 + 1 X − T  , so plugging the previously obtained form ulas for X t and T t in to (C9), w e get: ∂ x log ω 2 = X x 2  1 X + 1 X − 1 + 1 X − T  − 4( e 2 − e 1 ) X ( X − 1) X x +2 E 1 ( x + ω 3 ) − 2 E 1 ( ω 3 ) . (C10) 50 T o pro ceed, we substitute X = ϕ 2 1 ( x ) e 2 − e 1 , X − 1 = ϕ 2 2 ( x ) e 2 − e 1 , X − T = ϕ 2 3 ( x ) e 2 − e 1 and X x = − 2 ϕ 1 ( x ) ϕ 2 ( x ) ϕ 3 ( x ) e 2 − e 1 . This yields ∂ x log ω 2 = − ϕ 2 ( z ) ϕ 3 ( z ) ϕ 1 ( z ) − ϕ 1 ( z ) ϕ 3 ( z ) ϕ 2 ( z ) + ϕ 1 ( z ) ϕ 2 ( z ) ϕ 3 ( z ) + 2 E 1 ( x + ω 3 ) − 2 E 1 ( ω 3 ) . (C11) The ﬁnal result obta ined with the help of (B2 8) is ∂ x log ω 2 = − E 1 ( x ) + 3 X j =1 h E 1 ( x + ω j ) − E 1 ( ω j ) i , (C12) or, in the inte grated form, ω 2 ( x, t ) = ϑ 2 ( x ) ϑ 3 ( x ) ϑ 0 ( x ) ϑ 1 ( x ) g ( y , t ) , (C13) where the function g ( y , t ) can not b e ﬁxed b y the ab ov e argumen ts. Using the iden tit y 2 ϑ ′ 1 (0) ϑ 0 (0) ϑ 2 (0) ϑ 3 (0) ϑ 2 ( x ) ϑ 3 ( x ) ϑ 1 ( x ) ϑ 0 ( x ) = − ℘ ′ ( x ) ℘ ( x ) − e 3 , w e can express ω 2 in terms of the ℘ -function: ω 2 ( x, t ) = ℘ ′ ( x ) θ 2 0 ( x ) 2( ℘ ( x ) − e 3 ) ρ 2 ( t ) (C14) with some ρ ( t ) to b e determined from the condition that the x -indep enden t part of the p oten tial in the non-stationary Schr¨ odinger equation b e equal to the classical Hamiltonian H VI ( ˙ u, u ). It is the form (C14) tha t is more conv enien t to use in Section 8.3. References [1] P .P ainlev ´ e, Memoir e sur les ´ equations diﬀ´ er entiel le s don t l’int´ egr ale g ´ en ´ er ale est uniforme , Bull. So c. Math. Ph ys. F r a nce 28 (19 0 0) 20 1-261; P .P ainlev ´ e, Sur les ´ equations diﬀ´ er e n tiel les du se c ond or dr e et d ’ o r dr e sup´ erieur d ont l’int´ egr ale g´ en ´ er ale est uniforme , Acta Math. 21 (190 2 ) 1-85 [2] R.F uchs, Sur quelques ´ equations diﬀ´ er en tiel les lin´ ea r es du se c ond or dr e , C. 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